convergence and summability of fourier transforms and hardy spaces

Applied and Numerical Harmonic Analysis

Ferenc Weisz

Convergence and Summability of Fourier Transforms and Hardy Spaces

Applied and Numerical Harmonic Analysis

Series EditorJohn J. BenedettoUniversity of MarylandCollege Park, MD, USA

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Akram AldroubiVanderbilt UniversityNashville, TN, USA

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Stéphane JaffardUniversity of Paris XIIParis, France

Jelena KovacevicCarnegie Mellon UniversityPittsburgh, PA, USA

Gitta KutyniokTechnische Universität BerlinBerlin, Germany

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More information about this series at http://www.springer.com/series/4968

http://www.springer.com/series/4968

Ferenc Weisz

Convergence and Summabilityof Fourier Transformsand Hardy Spaces

Ferenc WeiszDepartment of Numerical AnalysisERotvRos Loránd UniversityBudapest, Hungary

ISSN 2296-5009 ISSN 2296-5017 (electronic)Applied and Numerical Harmonic AnalysisISBN 978-3-319-56813-3 ISBN 978-3-319-56814-0 (eBook)DOI 10.1007/978-3-319-56814-0

Library of Congress Control Number: 2017951129

Mathematics Subject Classification (2010): 42B08, 42A38, 42B30

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To Mártifor her patienceand love

Preface

The main purpose of this book is to investigate the convergence and summabilityboth of one-dimensional and multi-dimensional Fourier transforms.

It is known that the Fourier transform of f 2 L1.R/ is given by

bf .x/ D 1p2�

Z

R

f .u/e�{xu du .x 2 R/;

where { D p�1. If f 2 Lp.R/ for some 1 � p � 2 andbf 2 L1.R/, then the Fourierinversion formula holds:

f .x/ D 1p2�

Z

R

bf .u/e{xu du .x 2 R/:

In other cases, we introduce the Dirichlet integrals stf by:

sT f .x/ D 1p2�

Z T

�T

bf .u/e{xu du:

One of the deepest results in harmonic analysis is Carleson’s theorem [52, 187], i.e.for f 2 Lp.R/, 1 < p < 1,

limT!1 sT f D f a.e.

The convergence holds also in the Lp.R/-norm. In this book, we do not proveCarleson’s theorem as it is investigated exhaustively in several books (e.g. Ariasde Reyna [8] or Grafakos [152] or Muscalu and Schlag [253]).

This convergence does not hold for p D 1. However, using a summabilitymethod, say the Fejér method, we can generalize these results. The most knownresult in summability theory is Lebesgue’s theorem [212] about the Fejér means

vii

viii Preface

[116], i.e. the Fejér means of an integrable function converge almost everywhere tothe function:

limT!1

1

T

Z T

0

st f .x/ dt D f .x/ a.e.

The set of convergence was characterized as the Lebesgue points of f .In this book, these results will be proved and generalized for one-dimensional

and multi-dimensional Fourier transforms.The book is structured as follows. At the beginning of each chapter a brief survey

is given. A relatively new application of distribution theory is dealt with. Moreexactly, the theory of one- and multi-dimensional Hardy spaces is applied in Fourieranalysis. In Chap. 1, one-dimensional Hardy spaces are discussed. Some inequalitiesfor the Hardy-Littlewood maximal operator and the atomic decomposition of theHardy spaces are verified. Then the interpolation spaces of the Hardy spaces arecharacterized. Using the atomic decomposition, we give a sufficient condition foran operator such that it is bounded from the Hardy space to Lp.

In Chap. 2, one-dimensional Fourier transforms are considered. Some basic factsabout Fourier transforms and tempered distributions are given and the Fourierinversion formula is shown. We take a general summability method, the so-called�-summation defined by a function � W R ! R. This summation contains allwell-known summability methods, such as the Fejér, Riesz, Weierstrass, Abel,Picard, Bessel, Rogosinski, de La Vallée-Poussin summations. We prove that themaximal operators of the summability means are bounded from the Hardy spaceHp to Lp, whenever p > p0 for some p0 < 1. The critical index p0 dependson the summability method. For p D 1, we obtain a weak type inequality byinterpolation which implies the almost everywhere convergence of the summabilitymeans. The one-dimensional version of the almost everywhere convergence andthe weak type inequality are proved usually with the help of a Calderon-Zygmundtype decomposition lemma. However, in higher dimensions, this lemma cannot beused for all cases investigated in this monograph. Our method, which can also beapplied efficiently in higher dimensions, can be regarded as a new method to provethe almost everywhere convergence and weak type inequalities. The convergencetheorem about Lebesgue points mentioned above will be proved as well. Finally,strong summability will be considered. Using the modern techniques of two- andmulti-dimensional summability theorems, we give simple proofs for the strongsummability results later in Chap. 5. After the classical books of Bary [16] andZygmund [400], this is the first book which considers strong summability. Ourmethod is very different from that of Zygmund and Bary. At the end of this chaptersome summability methods are presented as special cases of the �-summation.

In Chap. 3, different types of Hardy-Littlewood maximal operators and multi-dimensional Hardy spaces (denoted by H�

p and Hp) are introduced. The methodsof proofs for one and several dimensions are entirely different; in most cases, thetheorems stated for several dimensions are much more difficult to verify. The atomicdecomposition of each Hardy space and the interpolation spaces between these

Preface ix

Hardy spaces are verified. Sufficient conditions for an operator to be bounded fromthe Hardy space to Lp are given for each Hardy space. It is very interesting that inthis result not only the one- and two-dimensional cases are different, but there isalso an essential difference between the two- and multi-dimensional cases.

In Chap. 4, some simple facts about multi-dimensional Fourier transformsare mentioned and the norm and almost everywhere convergence of the multi-dimensional Dirichlet integrals are verified. In the next chapter, we will considerdifferent summation methods for multi-dimensional trigonometric Fourier trans-forms. Basically, two types of summations will be introduced. In the first one, wetake the integral in the summability means over the balls of `q and call it as `q-summability. In the literature, the cases q D 1; 2;1, i.e. the triangular, circularand cubic summability, are investigated. In the second version of summation,which is investigated in Chap. 6, we take the integrals over rectangles which iscalled rectangular summability. The �-summability is considered in each version. InChap. 5, it will be proved that the maximal operators of the `q-summability meansare bounded from H�

p to Lp, whenever p > p0 for some p0 < 1. Here the criticalindex p0 depends on the summability method and the dimension. As we mentionedbefore, for p D 1, we obtain a weak type inequality by interpolation in this case, too,which implies again the almost everywhere convergence of the summability means.In two small sections, we will present some results about the circular Bochner-Riesz summability below the critical index, the proofs of which can be found in thebooks of Grafakos [152, 154, 155] and Lu and Yan [239]. Finally, new Lebesguepoints are introduced and the convergence at these Lebesgue points is proved forfunctions from the Wiener amalgam spaces. One of the novelties of this book is thatthe Lebesgue points are studied also in the theory of multi-dimensional summability.

In the last chapter, rectangular �-summability is investigated and similar resultsare proved as for the `q-summability. In this case, two types of convergence andmaximal operators are considered, namely the restricted (convergence over thediagonal or more generally over a cone) and the unrestricted (convergence overR

d). We show that the maximal operators of the rectangular summability means arebounded from Hp to Lp, whenever p > p0 for some p0 < 1. This implies the almosteverywhere convergence of the summability means. The theorems about Lebesguepoints are formulated in this case, too.

This book was aimed to be written so that it is as nearly self-contained aspossible. However, it is assumed that the reader has some basic knowledge onanalysis and functional analysis. Besides the classical results, recent results of thelast 20–30 years are studied. I hope the book will be useful for researchers as wellas for graduate or postgraduate students. Especially the first two chapters can beused well by graduate students and the other chapters rather by PhD students andresearchers.

Budapest, Hungary Ferenc Weisz

Acknowledgments

I am very grateful for the special atmosphere I feel amongst my colleagues at EötvösLoránd University, Budapest. Their friendship and professional knowledge inspiredme during my work a lot. My thanks are due to the Hungarian Scientific ResearchFunds (OTKA) No K115804 for supporting my research.

I would like to thank my colleague Péter Simon and my doctoral student, KristófSzarvas, for reading through the manuscript carefully and for their useful comments.I also thank Péter Kovács for his helpful assistance in creating the figures.

Above all, I am particularly indebted to my family, Márti, Ágoston, Gellért andAmbrus, the source of my happiness and inspiration. Their love and understandingare a continuous encouragement for me.

xi

Notations

.in; jn/, 2211H , 9B.c; h/, 138C.X/, 4C1.R/, 15C0.R/, 4Cc.R/, 4C1

c .R/, 15Cu.R/, 99DT , 91, 213Dq

T , 213Dn, 86, 207Dq

n, 205Ds, 73, 83D1

T;.il ;jl/, 221

Eq.R/, 101Eq.R

d/, 318E�q.R

d/, 393GT , 219Hc, 32Hi

p.Rd/, 152

Hp;1.R/, 22Hp;1.Rd/, 152H�

p;1.Rd/, 152Hp.R/, 22Hp.R

d/, 152H�

p .Rd/, 152

I.c; h/, 9K�

T , 96, 384

Kq;�T , 230

Km, 23, 153L0.R/, 13Lp.log L/k.Rd/, 138

Lp.T/, 4Lp.X/, 4L�p.R

d/, 393Lloc

p .R/, 4Lp;1.R/, 5M. p/, 158M! , 73, 83Mpf , 12, 139Ms f , 143Ms;pf , 150M f , 8N. p/, 23, 153P, 22PCf , 22PC, 87PC

�f , 152

P�, 87Pd , 152P5f , 22P5

�f , 152

Pt, 22Pd

t , 152S.R/, 16S.Rd/, 151S0.R/, 18S0.Rd/, 151S� f , 153S f , 153Sx, 159Tx, 73, 83U.1/r;p f , 320

U.2/r;p f , 321

Ur;pf , 320

xiii

xiv Notations

U.1/r f , 320

Urf , 320V� f , 183W.C; `q/.R/, 98W.C; `q/.R

d/, 138W.L1; `1/.R/, 98W.Lp.log L/k ; `q/.R

d/, 138W.Lp; `q/.R/, 98W.Lp; `q/.R

d/, 138W.Lp; c0/.R/, 98W.Lp; c0/.Rd/, 138W.Lp;1; `1/.R/, 106W.Lp;1; `q/.R/, 98W.Lp;1; `q/.R

d/, 138WI.Lp.log L/k ; `1/.Rd/, 143WI.Lp.log L/k ; c0/.Rd/, 143WI.Lp; `1/.Rd/, 142WI.Lp; c0/.Rd/, 143Œx1; : : : ; xn�f , 218PEq.R/, 102PEq.R

d/, 318PE�q.Rd/, 404PDp.I.0; 1//, 116PDp.R/, 107PD0

p.I.0; 1//, 117`p, 138`p.Z/, 4Of , 72, 77, 80, 81, 85, 203, 204Ou, 83{, 72�, 4logC u, 137F�1, 77F f , 72, 203I, 221Mf , 141, 142M.F/, 157M.1/

p f , 142M.1/f , 140, 142M.2/

p f , 140M1.F/, 157Mi.F/, 157Mpf , 142

M.1/p f , 140

M.2/p f , 140

Mpf , 141PN.p/, 33�Cf , 23, 153�

C

�f , 153

�5 f , 23, 153�

5

�f , 153

��

f , 103, 398

�q;��

f , 264��T f , 96, 97, 384

�q;�T f , 230, 234��

�f , 386

�, 23��

�f , 398

��T f , 398�0, 226, 230�.q/0 , 230

Qf , 87.x/, 153%.x;H/, 32_, 120^, 120ef , 87c0.Z/, 4f � g, 7f �, 73f _, 75, 203f�, 23, 153f]i , 152fm;�, 23, 153id, 220ms f , 147mpf , 105rI, 9rI.x; h/, 9sq

T f , 213, 217sT f , 91, 92, 213, 217sn f , 86sq

nf , 205snf , 207u � x, 138u_, 83C, 4N, 4P, 4Q, 4QC, 4R, 4RC, 4R

dC

, 183R

d! , 386

T, 4X, 4Z, 4soc , 120.A0;A1/�;q, 49Hp;q.R/, 48Jk , 223K.t; f ;A0;A1/, 48Lp;q.R/, 47

Notations xv

N. p/, 40, 156T� f , 13Tx, 20V� f , 63

Mf , 20Qf , 47sq�

f , 210, 217R

d! , 386

Contents

Part I One-Dimensional Hardy Spaces and Fourier Transforms

1 One-Dimensional Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 The Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Hardy-Littlewood Maximal Function.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Schwartz Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Tempered Distributions and Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5 Inequalities with Respect to Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.6 Atomic Decomposition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.7 Interpolation Between Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.8 Bounded Operators on Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2 One-Dimensional Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.2 Tempered Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.3 Partial Sums of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.4 Convergence of the Inverse Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 902.5 Summability of One-Dimensional Fourier Transforms .. . . . . . . . . . . . . 952.6 Norm Convergence of the Summability Means . . . . . . . . . . . . . . . . . . . . . . 982.7 Almost Everywhere Convergence of the Summability Means . . . . . . 1012.8 Boundedness of the Maximal Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082.9 Convergence at Lebesgue Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122.10 Strong Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.11 Some Summability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Part II Multi-Dimensional Hardy Spaces and Fourier Transforms

3 Multi-Dimensional Hardy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.1 Multi-Dimensional Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.1.1 Hardy-Littlewood Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . 1373.1.2 Strong Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

xvii

xviii Contents

3.2 Multi-Dimensional Tempered Distributions and Hardy Spaces . . . . . 1513.3 Inequalities with Respect to Multi-Dimensional Hardy Spaces . . . . . 1543.4 Atomic Decompositions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.4.1 Atomic Decomposition of H�p .R

d/. . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.4.2 Atomic Decomposition of Hp.R

d/ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.5 Interpolation Between Multi-Dimensional Hardy Spaces . . . . . . . . . . . 175

3.5.1 Interpolation Between the H�p .R

d/ Spaces . . . . . . . . . . . . . . . . . . 1753.5.2 Interpolation Between the Hp.R

d/ Spaces . . . . . . . . . . . . . . . . . . . 1763.6 Bounded Operators on Multi-Dimensional Hardy Spaces . . . . . . . . . . . 183

3.6.1 Bounded Operators on H�p .R

d/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1833.6.2 Bounded Operators on Hp.R

d/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

4 Multi-Dimensional Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.2 Multi-Dimensional Partial Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.3 Convergence of the Inverse Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 2134.4 Multi-Dimensional Dirichlet Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.4.1 Triangular Dirichlet Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.4.2 Circular Dirichlet Kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5 `q-Summability of Multi-Dimensional Fourier Transforms . . . . . . . . . . . . . 2295.1 The `q-Summability Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2295.2 Norm Convergence of the `q-Summability Means . . . . . . . . . . . . . . . . . . . 234

5.2.1 Proof of Theorem 5.2.1 for q D 1 and q D 1 . . . . . . . . . . . . . . . 2355.2.2 Some Summability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2575.2.3 Further Results for the Bochner-Riesz Means . . . . . . . . . . . . . . . 259

5.3 Almost Everywhere Convergence of the `q-Summability Means . . . 2645.3.1 Proof of Theorem 5.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2675.3.2 Proof of Theorem 5.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3095.3.3 Some Summability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.3.4 Further Results for the Bochner-Riesz Means . . . . . . . . . . . . . . . 313

5.4 Convergence at Lebesgue Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3175.4.1 Circular Summability .q D 2/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3175.4.2 Cubic and Triangular Summability (q D 1 and q D 1) . . . . 319

5.5 Proofs of the One-Dimensional Strong Summability Results . . . . . . . 374

6 Rectangular Summability of Multi-DimensionalFourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3836.1 Norm Convergence of Rectangular Summability Means . . . . . . . . . . . . 3836.2 Almost Everywhere Restricted Summability . . . . . . . . . . . . . . . . . . . . . . . . . 3866.3 Restricted Convergence at Lebesgue Points . . . . . . . . . . . . . . . . . . . . . . . . . . 3936.4 Almost Everywhere Unrestricted Summability . . . . . . . . . . . . . . . . . . . . . . 3986.5 Unrestricted Convergence at Lebesgue Points . . . . . . . . . . . . . . . . . . . . . . . 404

Contents xix

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Applied and Numerical Harmonic Analysis (80 Volumes) . . . . . . . . . . . . . . . . . . 429

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

List of Figures

Fig. 2.1 Dirichlet kernel DT for T D 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Fig. 2.2 Fejér kernel K�

T for T D 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Fig. 4.1 Regions of the `q-partial sums for d D 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Fig. 4.2 The Dirichlet kernel Dq

n with d D 2, q D 1, n D 4 . . . . . . . . . . . . . . . . . 206Fig. 4.3 The Dirichlet kernel Dq

n with d D 2, q D 2, n D 4 . . . . . . . . . . . . . . . . . 206Fig. 4.4 The Dirichlet kernel Dq

n with d D 2, q D 1, n D 4 . . . . . . . . . . . . . . . 207Fig. 4.5 The rectangular Dirichlet kernel with d D 2, n1 D 3, n2 D 5 . . . . . 208Fig. 4.6 The projections PC

1 PC2 , Q1 and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Fig. 4.7 The Dirichlet kernel DqT with d D 2, q D 1, T D 4 . . . . . . . . . . . . . . . . 214

Fig. 4.8 The Dirichlet kernel DqT with d D 2, q D 2, T D 4 . . . . . . . . . . . . . . . . 214

Fig. 4.9 The Dirichlet kernel DqT with d D 2, q D 1, T D 4 . . . . . . . . . . . . . . . 215

Fig. 4.10 The rectangular Dirichlet kernel with d D 2, T1 D 3, T2 D 5 . . . . . 215

Fig. 5.1 The Fejér kernel Kq;�T with d D 2, q D 1, T D 4 . . . . . . . . . . . . . . . . . . . 231

Fig. 5.2 The Fejér kernel Kq;�T with d D 2, q D 1, T D 4 . . . . . . . . . . . . . . . . . . 231

Fig. 5.3 The Fejér kernel Kq;�T with d D 2, q D 2, T D 4 . . . . . . . . . . . . . . . . . . . 232

Fig. 5.4 The sets Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Fig. 5.5 Weierstrass summability function �0.t/ D e�ktk22=2 . . . . . . . . . . . . . . . . . 257Fig. 5.6 Picard-Bessel summability function with d D 2 . . . . . . . . . . . . . . . . . . . 258Fig. 5.7 Riesz summability function with d D 2, ˛ D 1, D 2 . . . . . . . . . . . . 258Fig. 5.8 The Bochner-Riesz kernel K2;˛

T with d D 2, T D 4, ˛ D 1, D 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Fig. 5.9 The Bochner-Riesz kernel K2;˛T with d D 2, T D 4,

˛ D 1=10, D 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260Fig. 5.10 Unboundedness of �2;˛T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261Fig. 5.11 Boundedness of �2;˛T when d � 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261Fig. 5.12 Boundedness of �2;˛T when d D 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262Fig. 5.13 Boundedness of �2;˛T when d � 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262Fig. 5.14 Open question of the boundedness of �2;˛T when d � 3 . . . . . . . . . . . . 263

xxi

xxii List of Figures

Fig. 5.15 Unboundedness of �2;˛� from Lp.Rd/ to Lp;1.Rd/ . . . . . . . . . . . . . . . . . 313

Fig. 5.16 Boundedness of �2;˛� from Lp.R2/ to Lp;1.R2/ when d D 2 . . . . . . 314

Fig. 5.17 Boundedness of �2;˛� from Lp.Rd/ to Lp;1.Rd/ when d � 3 . . . . . . 315

Fig. 5.18 Open question of the boundedness of �2;˛� when d � 3 . . . . . . . . . . . . 315Fig. 5.19 Almost everywhere convergence of �2;˛T f , f 2 Lp.R

d/ . . . . . . . . . . . . . 316Fig. 5.20 The sets Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Fig. 6.1 The rectangular Fejér kernel K�T with d D 2, T1 D 3, T2 D 5 . . . . . 385

Fig. 6.2 The cone for d D 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Part IOne-Dimensional Hardy Spaces

and Fourier Transforms

Chapter 1One-Dimensional Hardy Spaces

The theory of the one-dimensional classical Hardy spaces is a very importanttopic of harmonic analysis and summability theory. In this chapter, we focus ourinvestigations on the atomic decomposition of the Hardy spaces. The Hardy spacesare investigated in many books, for example in Duren [93], Stein [308, 309],Stein and Weiss [311], Lu [233], Uchiyama [340] and Grafakos [152]. Beyondthese, the Hardy spaces have been introduced for martingales as well (see e.g.Garsia [127], Neveu [260], Dellacherie and Meyer [85, 86], Long [232] and Weisz[347]).

In Sect. 1.1, we introduce the Lp.R/ spaces and prove some basic inequalities.In Sect. 1.2, the Hardy-Littlewood maximal function is considered and we provethat it is bounded on the Lp.R/ spaces .1 < p � 1/ and is of weak type .1; 1/.The Lebesgue’s differentiation theorem is also proved. We introduce the Schwartzfunctions, tempered distributions, Hardy spaces and verify some inequalities forHardy spaces.

The atomic decompositions play an important role in this monograph. The firstversion of this decomposition can be found in Coifman and Weiss [67]. An atom isa simple, easy to handle function. The tempered distribution of the Hardy spaces isdecomposed into a sum of atoms. The advantage of this decomposition is that manytheorems need to be proved for atoms, only. In Sect. 1.6, we give a detailed prooffor the atomic decomposition of the Hardy spaces.

In the next section, the interpolation spaces between the Hardy spaces are given.These results are due to Fefferman et al. [109] (see also Weisz [355]). In the lastsection, we give a sufficient condition such that a sublinear operator is boundedfrom the Hardy space to Lp.R/.

© Springer International Publishing AG 2017F. Weisz, Convergence and Summability of Fourier Transformsand Hardy Spaces, Applied and Numerical Harmonic Analysis,DOI 10.1007/978-3-319-56814-0_1

3

4 1 One-Dimensional Hardy Spaces

1.1 The Lp Spaces

Let us denote the set of complex numbers, the set of real numbers, the set of rationalnumbers, the set of integers, the set of non-negative integers and the set of positiveintegers by C, R, Q, Z, N and P, respectively. The subsets of R and Q containingonly positive numbers are denoted by RC and QC, respectively.T denotes the torus,which can be identified naturally with the interval Œ��; �/.

In this book the constants C are absolute constants and the constants Cp (resp.Cp;q) are depending only on p (resp. p and q) and may denote different constants indifferent contexts.

The space Lp.X/ is consisting of all Lebesgue measurable functions f W X ! C,for which

k f kp WD�Z

X

j f jp d�

�1=p

; if 0 < p < 1

and

k f k1 WD supX

j f j; if p D 1;

where X � R is an arbitrary Lebesgue measurable set and � denotes the Lebesguemeasure. Two functions in Lp.X/ will be considered equal if they are equal �-almosteverywhere. It is known that Lp.X/ is a Banach space if 1 � p � 1 and a completequasi-normed space if 0 < p � 1. We also use the notation jIj for the Lebesguemeasure of the set I. Most often we will use the notation X D R or X D T. Lloc

p .R/

.1 � p < 1/ denotes the space of measurable functions f for which j f jp is locallyintegrable. Following from the definition of T, the functions from the Lp.T/ spacecan be extended to R such that they are periodic with respect to 2� . In case ofX D Z, the corresponding space will be denoted by `p.Z/ and it is consisting of allcomplex sequences c D .ck; k 2 Z/, for which

kck`p WD

X

k2Zjckjp

!1=p

; if 0 < p < 1

and

kck1 WD supk2Z

jckj ; if p D 1:

The subspace of `1.Z/ containing sequences vanishing at ˙1 is denoted by c0.Z/.The space of continuous functions with the supremum norm is denoted by C.X/

and Cc.R/ denotes the space of continuous functions having compact support. Wewill use the notation C0.R/ for the space of continuous functions vanishing at

1.1 The Lp Spaces 5

infinity, i.e.

C0.R/ WD�

f W R ! C W f 2 C.R/; limjxj!1

f .x/ D 0

�

:

We also introduce the notion of weak Lp.R/ spaces.

Definition 1.1.1 A measurable function f is in the weak Lp.R/ space, or, in otherwords, in the Lp;1.R/ .0 0

� �.j f j > �/1=p < 1: (1.1.1)

In case of p D 1, let Lp;1.R/ WD L1.R/.The following properties of the weak Lp.R/ spaces can be shown easily:

k f kp;1 D 0 ” f D 0 a.e.

kcf kp;1 D jcjk f kp;1 .c 2 C/;

k f C gkp;1 � cp.k f kp;1 C kgkp;1/;

where cp D max.2; 21=p/. Because of the last property k�kp;1 is a quasi-norm.We show that the weak Lp.R/ spaces are larger than the Lp.R/ spaces.

Proposition 1.1.2 If 0 �gj f .x/jp dx � �p� .j f j > �/ ;

which proves the proposition. �Note that the inclusion Lp.R/ � Lp;1.R/ is proper if 0 �� D 2�p��p D 2:

Recall that the weak space Lp;1.R/ is also complete for each p.Now we formulate Minkowski’s inequality for the Lp spaces.

Theorem 1.1.3 If 1 � p < 1 and f is a two-dimensional measurable function,then

�Z

R

ˇ

ˇ

ˇ

ˇ

Z

R

f .x; t/ dt

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

�Z

R

�Z

R

j f .x; t/jp dx

�1=p

dt;

whenever the right-hand side is finite.

Proof First we suppose that the left-hand side does exist. By the theorem about thedual spaces of Lp.R/, we have

khkp D supkgkp0 �1

ˇ

ˇ

ˇ

ˇ

Z

R

hg d�

ˇ

ˇ

ˇ

ˇ

;

where p0 is the conjugate index to p, i.e. 1p C 1p0

D 1. Then

�Z

R

ˇ

ˇ

ˇ

ˇ

Z

R

f .x; t/ dt

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

D

Z

R

f .�; t/ dt

p

D supkgkp0 �1

ˇ

ˇ

ˇ

ˇ

Z

R

�Z

R

f .x; t/ dt

�

g.x/ dx

ˇ

ˇ

ˇ

ˇ

� supkgkp0 �1

Z

R

Z

R

j f .x; t/j jg.x/j dx dt:

Applying Hölder’s inequality for the inner integral,

�Z

R

ˇ

ˇ

ˇ

ˇ

Z

R

f .x; t/ dt

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

� supkgkp0 �1

Z

R

�Z

R

j f .x; t/jp dx

�1=p �Z

R

jg.x/jp0

dx

�1=p0

dt

�Z

R

�Z

R

j f .x; t/jp dx

�1=p

dt;

which shows the result. Let us decompose f into positive and negative parts, f DfC C f�. We have just proved the inequality for both fC and f�. Thus the left-handside is finite for fC and f� as well as for f . Then the theorem follows from the firstpart of the proof. �Remark 1.1.4 Note that Minkowski’s inequality can also be written as

Z

R

f .�; t/ dt

p

�Z

R

k f .�; t/kp dt:

1.1 The Lp Spaces 7

Definition 1.1.5 The convolution of two functions f ; g 2 L1.R/ is defined by

. f � g/.x/ WDZ

R

f .x � t/g.t/ dt .x 2 R/:

It is easy to see that

. f � g/.x/ DZ

R

f .t/g.x � t/ dt .x 2 R/:

Of course,

k f � gk1 � k f k1kgk1:This inequality is generalized in the next Young’s inequality.

Theorem 1.1.6 Let f 2 Lr.R/, g 2 L1.R/ and 1 � r � 1. Then f � g 2 Lr.R/ and

k f � gkr � k f krkgk1:

Proof The inequality is trivial for r D 1. For 1 � r < 1,

�Z

R

j. f � g/.x/jr dx

�1=r

��Z

R

�Z

R

j f .x � t/jjg.t/j dt

�r

dx

�1=r

:

By Minkowski’s inequality,

k f � gkr �Z

R

�Z

R

.j f .x � t/jjg.t/j/r dx

�1=r

dt

DZ

R

jg.t/j�Z

R

j f .x � t/jr dx

�1=r

dt

D k f krkgk1;

which shows the inequality. �This theorem is clearly a special case of the next one if q D 1 and p D r.

Theorem 1.1.7 Let f 2 Lp.R/, g 2 Lq.R/ and 1 � p; q; r � 1 such that 1p C 1q D

1C 1r . Then f � g 2 Lr.R/ and

k f � gkr � k f kp kgkq :

Proof If 1 � p; q < 1, then

1

q0 C 1

rC 1

p0 D 1;p

q0 C p

rD 1;

q

rC q

p0 D 1;

where p0 resp. q0 denotes the conjugate indices to p resp. q. Applying Hölder’sinequality to the indices q0, r and p0, we conclude

j. f � g/.x/j �Z

R

j f .t/jp=q0�j f .t/jp=rjg.x � t/jq=r

� jg.x � t/jq=p0

dt

�

j f .t/jp=q0

q0

j f .t/jp=rjg.x � t/jq=r

r

jg.x � t/jq=p0

p0

D k f kp=q0

p

�Z

R

j f .t/jpjg.x � t/jq dt

�1=r

kgkq=p0

q :

By Fubini’s theorem,

k f � gkr � k f kp=q0

p kgkq=p0

q

�Z

R

j f .t/jpZ

R

jg.x � t/jq dx dt

�1=r

D k f kp=q0

p kgkq=p0

q kgkq=rq k f kp=r

p

D k f kp kgkq :

If p D 1, then r D 1 and q D 1. �

1.2 Hardy-Littlewood Maximal Function

In this section we will show that the Hardy-Littlewood maximal function is boundedon Lp.R/ for 1 < p � 1 and it is of weak type .1; 1/. Using this result we obtainthe Lebesgue’s theorem concerning the derivative of the integral function, whichwill be used later several times.

Definition 1.2.1 For a locally integrable function f 2 Lloc1 .R/ the Hardy-Littlewood

maximal function is defined with the equality

Mf .x/ WD supx2I

1

jIjZ

Ij f j d� .x 2 R/;

where the open interval I contains the point x.It is easy to see that Mf is measurable. Indeed, if

x 2 fMf > �g WD fx W Mf .x/ > �g ;

then there exists an open interval I, such that x 2 I and

1

jIjZ

Ij f j d� > �:

1.2 Hardy-Littlewood Maximal Function 9

In this case I � fMf > �g, hence fMf > �g is an open set, and so Mf is measurable.Let us denote by I.c; h/ .c 2 R; h > 0/ the interval with centre c and radius h:

I.c; h/ WD fx 2 R W jx � cj < hg:

Now we can define the centred maximal function,

Mcf .x/ WD suph>0

1

jI.x; h/jZ

I.x;h/j f j d� .x 2 R/:

Of course, Mcf � Mf . On the other hand, if x 2 I.y; h/, then I.y; h/ � I.x; 2h/ andso Mf � 2Mcf . Let rI.x; h/ WD I.x; rh/ for r > 0. First we verify the next coveringlemma.

Lemma 1.2.2 (Vitali Covering Lemma) Let be given finitely many open intervalsIj and let

E D[

j

Ij:

Then there exists a finite subcollection I1; : : : ; Im of disjoint intervals, such that

mX

kD1jIkj � jEj

3:

Proof Let I1 be an interval of the collection fIjg with maximal radius. Next chooseI2 to have maximal radius among the subcollection of intervals disjoint with I1.We continue this process until we can go no further. Then the intervals I1; : : : ; Im

are disjoint. Observe that 3Ik contains all intervals of the original collection thatintersect Ik .k D 1; : : : ;m/. From this it follows that [m

kD13Ik contains all intervalsfrom the original collection. Thus

jEj �ˇ

ˇ

ˇ

ˇ

ˇ

m[

kD13Ik

ˇ

ˇ

ˇ

ˇ

ˇ

�mX

kD1j3Ikj � 3

mX

kD1jIkj ;

which shows the lemma. �Before we consider the maximal function, we show the next lemma. The

�.j f j > �/ WD � .fx W j f .x/j > �g/

function is called distribution function, with the help of which we can compute theLp-norm as follows. Let us denote the characteristic function of a set H by 1H, i.e.

1H.x/ WD�

1; if x 2 HI0; if x … H:

Proposition 1.2.3 If 0 �/ d�:

Proof By Fubini’s theorem,

pZ 1

0

�p�1 �.j f j > �/ d� DZ 1

0

p �p�1Z

R

1fxWj f .x/j>�g.x/ dx d�

DZ

R

Z 1

0

p �p�11fxWj f .x/j>�g.x/ d� dx

DZ

R

Z j f .x/j

0

p �p�1 d�

!

dx

DZ

R

j f .x/jp dx;

which shows our proposition. �

Theorem 1.2.4 The maximal operator M is of weak type .1; 1/, i.e.

sup�>0

��.Mf > �/ � 3k f k1 . f 2 L1.R//: (1.2.1)

Moreover, if 1 �g be a compact subset. For each x 2 fMf > �g there existsan open interval Ix such that x 2 Ix and

� <1

jIxjZ

Ix

j f j d�: (1.2.3)

Since x 2 Ix, it is easy to see that

E �[

x2E

Ix � fMf > �g:

Thus the fIxg open intervals are covering the compact set E, so we can choose afinite collection of these intervals covering E. By Lemma 1.2.2 we can choose a

finite disjoint subcollection I1; : : : ; Im of this covering with

jEj � 3

mX

kD1jIkj:

Since each Ik satisfies (1.2.3), adding these inequalities we can see that

jEj < 3

�

mX

kD1

Z

Ik

j f j d� � 3

�

Z

fM f>�gj f j d�:

Taking the supremum over all compact sets E � fMf > �g, we conclude

�.Mf > �/ � 3

�

Z

fM f>�gj f j d� � 3

�

Z

R

j f j d�;

which gives exactly (1.2.1).For p D 1 obviously

1

jIjZ

Ij f j d� � k f k1 ;

and so

kMf k1 � k f k1 . f 2 L1.R//:

Now the theorem follows easily for 1 �=2I0; if j f .x/j � �=2:

Then

j f .x/j � j f�.x/j C �=2 and Mf � Mf� C �=2:

So

fMf > �g � fMf� > �=2g:

However, by (1.2.1),

�.Mf� > �=2/ � 6

�

f�

1D 6

�

Z

fj f j>�=2gj f j d� D 6

�

Z

R

j f j1fj f j>�=2g d�:

This means that

�.Mf > �/ � �.Mf� > �=2/ � 6

�

Z

R

j f j1fj f j>�=2g d�:

Proposition 1.2.3 and Fubini’s theorem imply

kMf kpp D p

Z 1

0

�p�1 �.jMf j > �/ d�

� 6pZ 1

0

�p�1 1�

Z

R

j f .x/j1fj f j>�=2g.x/ dx d�

D 6pZ

R

j f .x/jZ 1

0

�p�21fj f j>�=2g.x/ d� dx

D 6pZ

R

j f .x/jZ 2j f .x/j

0

�p�2 d� dx

D 3 � 2pp

p � 1

Z

R

j f jp d�;

which gives the proof of the theorem. �Inequality (1.2.2) does not hold for p D 1. Indeed, let

f .x/ WD8

<

:

0; if x … .0; 1=2/I1

x ln2 x; if x 2 .0; 1=2/:

Since the primitive function of 1

x ln2 xis � ln�1 x, so f 2 L1.R/. Now if 0 < x < 1=2,

then

Mf .x/ � 1

2x

Z x

0

j f j d� D 1

2x ln.1=x/:

This last function is not integrable on the interval .0; 1=2/ because its primitivefunction is � ln ln.1=x/. This implies immediately that Mf … L1.R/. Moreover, itcan be shown that Mf … L1.R/ for f 2 L1.R/, f ¤ 0 a.e.

We generalize the maximal function as follows. For 1 � p < 1 and f 2 Llocp .R/

let us define

Mp f .x/ WD supx2I

�

1

jIjZ

Ij f jp d�

�1=p

.x 2 R/:

Since Mpp f D M.j f jp/ for 1 � p < 1, the following result follows from

Theorem 1.2.4.

Corollary 1.2.5 If 1 � p < 1, then

sup�>0

��.Mpf > �/1=p � Ck f kp . f 2 Lp.R//:

Moreover, if p < r � 1, then

kMpf kr � Crk f kr . f 2 Lr.R//:

The next density theorem is due to Marcinkiewicz and Zygmund [244]. Thistheorem is fundamental for the proof of the almost everywhere convergence andit is similar to the Banach-Steinhaus theorem, which can be applied for normconvergences well.

Let L0.R/ denote the set of measurable functions f W R ! C and X � L0.R/.Let the linear operators

T;Tn W X ! L0.R/ .n 2 N/

be given. Moreover set

T� f .x/ WD supn2N

jTn f .x/j . f 2 X; x 2 R/:

The operator T� is called the maximal operator of the sequence of operators .Tn; n 2N/.

Theorem 1.2.6 Let X be a normed space of measurable functions and S � X bedense in X. Suppose that

T f D limn!1 Tn f a.e.

for all f 2 S. If

sup�>0

� �.jT f j > �/ � Ck f kX . f 2 X/ (1.2.4)

and

sup�>0

� �.T� f > �/ � Ck f kX . f 2 X/; (1.2.5)

then for every f 2 X,

T f D limn!1 Tn f a.e.


Proof Fix f 2 X and set

� WD lim supn!1

jTn f � T f j:

It is sufficient to show that � D 0 a.e.Choose a sequence fm 2 S .m 2 N/ such that

limm!1 k f � fmkX D 0:

By the triangle inequality,

� � lim supn!1

jTn. f � fm/j C lim supn!1

jTn fm � T fmj C jT. fm � f /j

for all m 2 N. Since fm 2 S, we have

lim supn!1

jTn fm � T fmj D limn!1 jTn fm � T fmj D 0 a.e.;

so

� � T�. fm � f /C jT. fm � f /j a.e.

Applying the inequalities (1.2.4) and (1.2.5), we obtain

�.� > 2�/ � �.T�. fm � f / > �/C �.jT. fm � f /j > �/� C��1k fm � f kX C C��1k fm � f kX

for all � > 0 and m 2 N. Since fm ! f in the X-norm as m ! 1, we get that

�.� > 2�/ D 0

for all � > 0. This implies immediately that � D 0 almost everywhere. �Now we can state the Lebesgue’s differentiation theorem, which says that the

derivative of the integral of a function f is almost everywhere equal to f .

Corollary 1.2.7 Let rn > 0 .n 2 N/, limn!1 rn D 0 and f 2 Lloc1 .R/. Then

limn!1

1

2rn

Z xCrn

x�rn

f .t/ dt D f .x/ a.e. x 2 R:

Proof First of all, f is integrable on every compact set, thus the left-hand side iswell defined and we can suppose that f 2 L1.R/. Let

T f WD f and Tn f WD 1

2rn

Z xCrn

x�rn

f .t/ dt:

1.3 Schwartz Functions 15

These operators are linear and

sup�>0

� �.jT f j > �/ D sup�>0

� �.j f j > �/ � sup�>0

Z

fj f j>�gj f j d� � k f k1

implies (1.2.4). Inequality (1.2.5) follows from Theorem 1.2.4. Denote by S the setof continuous functions. If f 2 S, then the result obviously holds. Since S is dense inL1.R/, Theorem 1.2.6 implies the corollary for all f 2 L1.R/ and so for all locallyintegrable functions. �

The following corollary is an easy consequence.

Corollary 1.2.8 For all f 2 Lloc1 .R/,

limr!0

1

2r

Z xCr

x�rf .t/ dt D f .x/ a.e. x 2 R:

Of course we can take the integral only on one side of x: for all f 2 Lloc1 .R/,

limr!0

1

r

Z xCr

xf .t/ dt D f .x/ a.e. x 2 R:

This implies easily the next version of the result.

Corollary 1.2.9 For all f 2 Lloc1 .R/,

limx2I;jIj!0

1

jIjZ

If d� D f .x/ a.e. x 2 R:

Note that the last three corollaries hold also for all f 2 Lp.R/ .1 � p � 1/. Thecorollary implies that j f .x/j � Mf .x/ for almost every x 2 R, and so the converseof (1.2.2) is also true:

k f kp � kMf kp .1 � p � 1/:

1.3 Schwartz Functions

Let us denote by C1.R/ the set of infinitely many times differentiable functionsf W R ! C and let

C1c .R/ WD f f 2 C1.R/ W f has compact supportg :

Definition 1.3.1 The function f 2 C1.R/ is called a Schwartz function if for all˛; ˇ 2 N,

supx2R

ˇ

ˇx˛f .ˇ/.x/ˇ

ˇ D C˛;ˇ < 1:

The class of Schwartz functions will be denoted by S.R/. Now let us see someexamples for Schwartz functions. Obviously,

C1c .R/ � S.R/:


f .x/ D e�x2 2 S.R/:

However

f .x/ D e�jxj … S.R/;

since it is not differentiable at the point 0. Moreover,

f .x/ D

1C jxj4��a … S.R/ .a > 0/:

The next proposition follows easily from the definition.

Proposition 1.3.2 Let f 2 C1.R/. Then f 2 S.R/ if and only if for all ˛;N 2 N

there exists a constant C˛;N such that for all x 2 R

ˇ

ˇ f .˛/.x/ˇ

ˇ � C˛;N .1C jxj/�N :

The class S.R/ is not a normed space, but we could define a topology on it. Herewe introduce a convergence on S.R/.

Definition 1.3.3 Let fk; f 2 S.R/. We say that

fk ! f in S.R/

if for all ˛; ˇ 2 N,

supx2R

ˇ

ˇ

ˇx˛ . fk � f /.ˇ/ .x/ˇ

ˇ

ˇ ! 0; as k ! 1:

The convergence in S.R/ is stronger than the convergence in Lp.R/.

1.3 Schwartz Functions 17

Theorem 1.3.4 Let fk; f 2 S.R/ and 1 � p � 1. Then S.R/ � Lp.R/ and

f .ˇ/

p� Cp

f .ˇ/

1 C Cp supx2R

ˇ

ˇ

ˇ

ˇ

xj

2p

k

C1f .ˇ/.x/ˇ

ˇ

ˇ

ˇ

;

where ˇ 2 N and bxc denotes the integer part of x 2 R. If

fk ! f in S.R/; then fk ! f in Lp.R/:

Proof The inequality to be proved is clear for p D 1. If 1 � p < 1, then

f .ˇ/

pD�Z

jxj�1

ˇ

ˇ f .ˇ/.x/ˇ

ˇ

pdx C

Z

jxj>1jxj2 ˇˇ f .ˇ/.x/

ˇ

ˇ

p jxj�2 dx

�1=p

�

2

f .ˇ/

p

1 C

supjxj>1

ˇ

ˇ

ˇx2p f .ˇ/.x/

ˇ

ˇ

ˇ

p!

Z

jxj>1jxj�2 dx

!1=p

:

Since

.a C b/˛ � a˛ C b˛

for a; b � 0, 0 < ˛ � 1 and

Z 1

1

jxj�2 dx D 1;

we have

f .ˇ/

p�

2

f .ˇ/

p

1 C 2

supjxj>1

ˇ

ˇ

ˇx2p f .ˇ/.x/

ˇ

ˇ

ˇ

p!!1=p

� 21=p

f .ˇ/

1 C 21=p supjxj>1

ˇ

ˇ

ˇ

ˇ

xj

2p

k

C1f .ˇ/.x/ˇ

ˇ

ˇ

ˇ

:

For the proof of the convergence in the Lp.R/-norm, we apply the last inequalityfor f � fk:

. f � fk/.ˇ/

p� Cp

. f � fk/.ˇ/

1 C Cp supx2R

ˇ

ˇ

ˇ

ˇ

xj

2p

k

C1. f � fk/

.ˇ/ .x/

ˇ

ˇ

ˇ

ˇ

:

With the choice ˛ Dj

2p

k

C 1 and ˇ D 0 we obtain

k f � fkkp � Cp k f � fkk1 C Cp supx2R

jx˛. f � fk/.x/j :


Here the right-hand side tends to 0 because fk ! f in S.R/. Hence fk ! f in theLp.R/-norm as well. �

1.4 Tempered Distributions and Hardy Spaces

Definition 1.4.1 A map u W S.R/ ! C is called tempered distribution if it is linearand continuous, more exactly,

(a) u.˛1 f1 C ˛2 f2/ D ˛1u. f1/C ˛2u. f2/ for all f1; f2 2 S.R/ and ˛1; ˛2 2 C,(b) for all sequences . fk; k 2 N/ � S.R/ for which fk ! f in S.R/, one has

u. fk/ ! u. f /, as k ! 1.

Let us denote the set of tempered distributions by S0.R/. In other words, theelements of the dual space of S.R/, which is denoted by S�.R/, are called tempereddistributions, i.e. S0.R/ D S�.R/. Now let us present some examples for tempereddistributions. The linearity is in all examples trivial.

Example 1.4.2

(a) The most simple tempered distribution is the Dirac measure. Let

ı0. f / WD f .0/:

This is indeed a tempered distribution, since fk ! f in S.R/ implies that fk ! fin the L1.R/-norm. As fk is continuous, fk ! f uniformly and so everywhere.Thus

ı0. fk/ D fk.0/ ! f .0/ D ı0. f /:

(b) Let g 2 Lp.R/ for some 1 � p � 1 and

Lg. f / WDZ 1

�1f .x/g.x/ dx . f 2 S.R// :

By Hölder’s inequality the integral is well defined because f 2 S.R/ � Lp0.R/,1p C 1

p0D 1. If fk ! f in S.R/, then fk ! f in Lp0.R/ as well. Applying again

Hölder’s inequality,

ˇ

ˇLg. fk/� Lg. f /ˇ

ˇ �Z 1

�1j fk.x/ � f .x/j jg.x/j dx

� k fk � f kp0 kgkp ! 0;

as k ! 1, so Lg is a tempered distribution, indeed. So each function g 2Lp.R/ characterizes a tempered distribution. The function g and the generated

1.4 Tempered Distributions and Hardy Spaces 19

tempered distribution Lg will be considered identical. Thus in case g 2 Lp.R/

for some 1 � p � 1, we can say that g 2 S0.R/.(c) There exists a tempered distribution which is not an element of Lp.R/ .1 � p �

1/. Let

g.x/ WD .1C jxj/r .r 2 R/:

It is easy to see that in case r > 0, g 62 Lp.R/ .1 � p � 1/. However, Lg or gis a tempered distribution. Indeed, choosing m such that r � m < �1, we have

jLg. fk/ � Lg. f /j �Z

j fk.x/ � f .x/j jg.x/j dx

� supx2R

.1C jxj/m j fk.x/ � f .x/j�

Z 1

�1.1C jxj/r�m dx

� C supx2R

.1C jxj/m j fk.x/ � f .x/j�

! 0;

as k ! 1 and fk ! f in S.R/.(d) There exist tempered distributions which are not functions. Let be a finite

signed Borel measure and

L . f / WDZ 1

�1f d . f 2 S.R// :

If fk ! f in S.R/, then

ˇ

ˇL . fk/ � L . f /ˇ

ˇ �Z 1

�1j fk � f j dj j � k fk � f k1 j j.R/;

where j j denotes the total variation of . That is to say L is a tempereddistribution, so each finite Borel measure characterizes a tempered distribution.

The convergence for tempered distributions is introduced as follows.

Definition 1.4.3 Let u; uk 2 S0.R/ .k 2 N/ be tempered distributions. Then uk ! uin S0.R/ if uk. f / ! u. f / for all f 2 S.R/.

Recall that convergence in S.R/ implies convergence in Lp.R/ for all 1 � p � 1(see Theorem 1.3.4). Now we show that convergence in Lp.R/ implies convergencein S0.R/.

Theorem 1.4.4 If uk; u 2 Lp.R/ for some 1 � p � 1 and uk ! u in the Lp.R/-norm, then uk ! u in S0.R/, as k ! 1.


Proof If f 2 S.R/, then

juk. f /� u. f /j Dˇ

ˇ

ˇ

ˇ

Z

R

f .x/ .uk.x/ � u.x// dx

ˇ

ˇ

ˇ

ˇ

�Z

R

j f .x/j juk.x/ � u.x/j dx

� k f kp0 kuk � ukp ! 0;

as k ! 1 because f 2 Lp0.R/. �The notions of product and convolution can easily be extended to tempered

distributions.

Definition 1.4.5 Let u 2 S0.R/ and f 2 S.R/. The product fu is given by

. fu/ .h/ WD u . f h/ .h 2 S.R// :

The product fu is well defined because f h is a Schwartz function. We introducethe reflection and translation operators by

Mf .x/ WD f .�x/; Tx f .t/ WD f .t � x/:

By Theorem 1.1.6 the convolution is well defined for all g 2 L1.R/ and f 2Lp.R/ .1 � p � 1/. Notice that for f ; g; h 2 S.R/,

Z

R

. f � g/ .x/h.x/ dx DZ

R

Z

R

f .t/g.x � t/h.x/ dt dx

DZ

R

f .t/

�Z

R

Mg.t � x/h.x/ dx

�

dt

DZ

R

f .t/ .Mg � h/ .t/ dt:

Keeping this property, we extend the convolution to tempered distributions.

Definition 1.4.6 Let u 2 S0.R/ and g 2 S.R/. The convolution u � g is defined by

.u � g/ .h/ WD u .Mg � h/ .h 2 S.R// :

The convolution is well defined because Mg�h 2 S.R/. Indeed, using some resultsabout Fourier transforms described later in Chap. 2, we know that bMg � h D bMgbf 2S.R/, sincebMg andbh and their product are Schwartz functions. The inverse Fouriertransform of a Schwartz function is a Schwartz function again, so Mg � h 2 S.R/.

Theorem 1.4.7 If u 2 S0.R/ and g 2 S.R/, then u � g is a C1 function.


Proof For h 2 S.R/ we have

u � g.h/ D u.Mg � h/ D u

�Z

R

Mg.� � x/h.x/ dx

�

D u

�Z

R

Tx Mg.�/h.x/ dx

�

:

The Riemann sums of the last integral are easily shown to converge in the topologyof S.R/. Since u is continuous,

u � g.h/ DZ

R

u.Tx Mg/h.x/ dx:

Thus

u � g.x/ D u.Tx Mg/: (1.4.1)

By Taylor’s formula

limh!0

Mg.x C h/� Mg.x/h

D Mg 0.x/ in S.R/;

hence

u � g.x C h/� u � g.x/

hD u

�

TxCh Mg � Tx Mgh

�

! �u�

Tx Mg 0�

as h ! 0. The same calculation for higher order derivatives completes the proof. �One can show that u � g is a tempered distribution. We can see easily that

Z

R

. f � g/ .x/h.x/ dx DZ

R

Z

R

f .x � t/g.t/h.x/ dx dt

DZ

R

Z

R

f .y/g.t/h.y C t/ dy dt

DZ

R

. f � Mh/.t/Mg.t/ dt

for all f ; g; h 2 S.R/.

Definition 1.4.8 A tempered distribution u 2 S0.R/ is said to be bounded if u � h 2L1.R/ for all h 2 S.R/. If u is a bounded tempered distribution and g 2 L1.R/,then let

u � g.h/ WD hu � Mh; Mgi DZ

R

.u � Mh/.x/Mg.x/ dx .h 2 S.R//:

The last integral is well defined, because u � Mh 2 L1.R/ and g 2 L1.R/. Notethat by Hölder’s inequality every function f 2 Lp.R/ .1 � p � 1/ is a boundedtempered distribution. This definition is the same as Definition 1.4.6 if g 2 S.R/.Indeed,

u � g.h/ D u � Mh.Mg/ D u.h � Mg/ D u.Mg � h/:

For a function � on R let

�t.x/ WD t�1�.x=t/ .t > 0/:

Let us introduce the operators, with the help of which the Hardy spaces can bedefined. Let

P.x/ WD P1.x/ WD 1

�.1C jxj2/ ;

Pt.x/ D t�1P.x=t/ D t

�.t2 C jxj2/ .t > 0; x 2 R/

be the Poisson kernel. Note thatR

RP.x/ dx D 1. For a bounded tempered

distribution f 2 S0.R/ let

P5f .x/ WD supt>0

supy2RWjx�yj<t

j. f � Pt/.y/j .x 2 R/;

PCf .x/ WD supt>0

j. f � Pt/.x/j .x 2 R/:

Since Pt is integrable, f � Pt is well defined for a bounded tempered distribution f .We claim that it is a bounded C1 function.

Proposition 1.4.9 For a bounded tempered distribution f , f � Pt is a bounded C1function.

Proof We can write P D � � h C with �; 2 S.R/ and h 2 L1.R/. To seethis, we use the Fourier transform of P,bP.�/ D e�j�j=

p2� . Let h D P, � 2 S.R/

such thatb�.�/ D 1 near to 0 and b .�/ D .1 �b�.�//e�j�j=p2� 2 S.R/. Therefore

bP D b� �bh C b which implies Pt D �t � ht C t and

f � Pt D . f � �t/ � ht C f � t;

which shows that f � Pt is a bounded C1 function. �

Definition 1.4.10 For 0 < p < 1 the Hardy spaces Hp.R/ and weak Hardy spacesHp;1.R/ consist of all bounded tempered distributions for which

k f kHpWD kP5f kp < 1 and k f kHp;1

WD kP5f kp;1 < 1:

For p D 1 let H1.R/ WD L1.R/. The Hardy spaces can also be defined withthe help of PC. There are also other equivalent norms on Hp.R/. For � 2 S.R/ withR

R� d� ¤ 0 let

�5f .x/ WD supt>0

supy2RWjx�yj<t

j. f � �t/.y/j .x 2 R/;

�Cf .x/ WD supt>0

j. f � �t/.x/j .x 2 R/:

P5f or �5f are called the non-tangential maximal function of f . Let m 2 P,

N.p/ WD b1=p � 1c; m > N.p/; � 2 S.R/

and

k�kKm WD supx2R;˛�m

.1C jxj/mC1j�.˛/.x/j:

Recall that bxc denotes the integer part of x 2 R. We say that a function � 2 S.R/is in the space Km when k�kKm < 1. The grand maximal function is defined by

f�.x/ WD fm;�.x/ WD supk�kKm �1

�5. f /.x/:

The next theorem says that for a tempered distribution f , fm;� 2 Lp.R/ if andonly if f is a bounded tempered distribution and f 2 Hp.R/.

Theorem 1.4.11 A tempered distribution f is in Hp.R/ .0 N.p/.

Note that � denotes the equivalence of norms and spaces, more exactly we writethat A � B if there exist positive constants c1 and c2 such that c1A � B � c2A. Aswe can see the Hardy spaces are independent of the choice of the functions � and ofm. The inequalities in (1.4.2) are well-known facts of the theory of Hardy spaces andthey can be found in several books and papers (e.g. in Stein [309], Grafakos [152],Lu [233], Stein [308], Stein and Weiss [311], Uchiyama [340], Fefferman and Stein[108], Fefferman, Riviere and Sagher [109, 274] and Wilson [383]), so we do notprove this theorem here.

We have seen in Theorem 1.4.4 that convergence in Lp.R/ .1 � p � 1/ isstronger than convergence in S0.R/. Now we show the same for the Hardy spaces.

Theorem 1.4.12 If fk; f 2 Hp.R/ for some 0 < p < 1 and fk ! f in the Hp.R/-norm, then fk ! f in S0.R/, as k ! 1.

Proof Using (1.4.1) we conclude

ˇ

ˇ

ˇ f . M�/ˇ

ˇ

ˇ D j f � �.0/j �

supx2R;˛�m

.1C jxj/mC1 ˇˇ�.˛/.x/

ˇ

ˇ

!

fm;�.z/;

where jzj < 1, m > N.p/ and � 2 S.R/. Then

ˇ

ˇ

ˇ f . M�/ˇ

ˇ

ˇ � C� infjzj<1

fm;�.z/ � C�

Z

R

f pm;�.z/ dz � C� k f kp

Hp;

which implies the convergence in the sense of tempered distributions. �The space L1.R/ \ Hp.R/ and the space L2.R/ \ Hp.R/ are dense in Hp.R/ for

all 0 < p � 1.

Theorem 1.4.13 If 0 < p < 1, then the space Lr.R/ \ Hp.R/ is dense in Hp.R/

for any p � r � 1.

Proof First assume that 0 < p � 1. We will show that f � Pt 2 Lr.R/ \ Hp.R/ forall p � r � 1 and f � Pt ! f in the Hp.R/-norm as t ! 0. To this end let � be asuitable smooth function with compact support such that

�.x/ D�

1; if jxj < 1I0; if jxj > 2:

For all x 2 R,

P.x/ D P.x/�.x/C1X

kD1

�.2�kx/P.x/� �.2�.k�1/x/P.x/�

D P.x/�.x/C1X

kD12�2k �.2

�kx/ � �.2�.k�1/x/�.2�2k C j2�kxj2/

D P.x/�.x/C1X

kD12�k .�k/2k .x/;

where

�k WD �.x/� �.2x/

�.2�2k C jxj2/ and .�k/2k .x/ WD 2�k�k.x=2k/:

Since

1

.2�2k C jxj2/j � 1

jxj2j.j 2 N/ ;

there is a positive constant c0 such that c0�P 2 Km and c0�k 2 Km for all k D1; 2; : : :. Then

j f � Pt.x/j � 1

c0j f � .c0�P/t .x/j C 1

c0

1X

kD12�k . f � .c0�k/2kt/ .x/;

which implies

j f � Pt.y/j � P5f .x/ � Cf�.x/; (1.4.3)

whenever jx�yj < t. Note that the first inequality follows directly from the definitionof P5f . Integrating over I.y; t/ with respect to x, we have

2t j f � Pt.y/jp � Cp

Z

I.y;t/f p�.x/ dx � Cp k f kp

Hp: (1.4.4)

From this it follows that f � Pt 2 L1.R/. On the other hand f � Pt 2 Lp.R/ becausef 2 Hp.R/. Therefore f � Pt 2 Lr.R/ for all p � r � 1.

We have to show yet that f � Pt 2 Hp.R/ and f � Pt ! f in the Hp.R/-normas t ! 0. For the first statement we use the formula Ps � Pt D PsCt. This is aconsequence of the fact that bPt.�/ D e�tj�j=

p2� by applying the Fourier transform.

Then

sups>0

jPs � Pt � f j D sups>0

jPsCt � f j � sups>0

jPs � f j

and

kPt � f kHp� k f kHp

.t > 0/ :

For the second statement, i.e. for the convergence in the Hp.R/-norm observe thatby (1.4.3),

supt>0

supy2RWjx�yj<t

j. f � Pt/.y/j < 1 for a.e. x 2 R:

It follows from Proposition 1.4.9 that f � Pt is a harmonic function. One can provethat these two statements imply the existence of the limit limt!0. f � Pt/.x/ foralmost every x 2 R (see e.g. Stein and Weiss [311, p. 64]). On the other hand,by (1.4.4),

limt!1. f � Pt/.x/ D 0 for all x 2 R:

Hence the function t 7! . f � Pt/.x/ is uniformly continuous for almost every x 2 R.In other words,

sups>0

jPs � Pt � f .x/ � Ps � f .x/j D sups>0

jPsCt � f .x/� Ps � f .x/j < �

if t is small enough, thus the supremum tends to 0, as t ! 0. Since

sups>0

jPs � Pt � f � Ps � f j D sups>0

jPsCt � f � Ps � f j � 2 sups>0

jPs � f j 2 Lp.R/;

the convergence

limt!0

Pt � f D f in the Hp.R/-norm

follows from the Lebesgue dominated convergence theorem. The theorem is obviousfor 1 < p < 1 since Hp.R/ � Lp.R/ in this case. �

Note that inequality (1.4.3) implies also that

kP5f kp � C k f�kp .0 1, Lp.R/ � Hp.R/.

Theorem 1.5.1 We have

k f kH1;1 D sup�>0

��.P5f > �/ � Ck f k1 . f 2 L1.R//: (1.5.1)

If 1 0 and x; y 2 R such that jx � yj < t. Clearly,

. f � Pt/.y/ D 1

t

Z

R

f .z/Py � z

t

�

dz:

Let E0.y/ WD fz W jy � zj < tg and Ek.y/ WD fz W 2k�1t � jy � zj < 2ktg .k � 1/.Then

j. f � Pt/.y/j � 1

t

1X

kD0

Z

Ek.y/j f .z/jP

y � z

t

�

dz

� C

t

1X

kD02�2k

Z

Ek.y/j f .z/j dz

1.5 Inequalities with Respect to Hardy Spaces 27

� C

t

1X

kD02�2k

Z

I.y;2kt/j f .z/j dz

� C1X

kD0

2�k

2kC1t

Z

I.x;2kC1t/j f .z/j dz

� CMf .x/:

Taking the supremum over all .y; t/, jx � yj < t, we get P5f � CMf .Inequalities (1.5.1) and (1.5.2) follow from Theorem 1.2.4. �

We will see from the next theorem that f ��t ! f in S.R/, in Lp.R/ and in S0.R/.

Theorem 1.5.2 Assume that � 2 S.R/ andR

R� d� D 1.

(a) If f 2 S.R/, then limt!0 f � �t D f in S.R/.(b) If f 2 Lp.R/ for some 1 � p < 1, then limt!0 f � �t D f in Lp.R/.(c) If f 2 S0.R/, then limt!0 f � �t D f in S0.R/.

Proof SinceR

R� d� D 1 and f 2 S.R/, we have

f � �t.x/� f .x/ DZ

R

�t.z/ . f .x � z/ � f .x// dz

DZ

R

�.y/ . f .x � yt/ � f .x// dy: (1.5.3)

By Lagrange’s mean value theorem,

supx2R

jx˛. f � �t.x/� f .x//j �Z

R

j�.y/j supx2R

jx˛ . f .x � yt/� f .x//j dy

�Z

R

j�.y/j jytj supx2R

ˇ

ˇx˛f 0.�/ˇ

ˇ dy;

where � D �.x/ 2 .x � yt; x/ and ˛ 2 N. We may suppose that x > 0 and 0 < t � 1.If yt > x=2, then

j�.y/j jytj sup0<x<2yt

ˇ

ˇx˛f 0.�/ˇ

ˇ � C j�.y/j jytj˛C1 supx2R

ˇ

ˇ f 0.�/ˇ

ˇ

� C j�.y/j jytj˛C1 ! 0;

as t ! 0. Obviously, the last term can be estimated by C j�.y/j jyj˛C1, which isintegrable because � 2 S.R/. If yt � x=2, then x � yt � x=2 and so � � x=2. Thus

j�.y/j jytj supx�2yt

ˇ

ˇx˛f 0.�/ˇ

ˇ � C j�.y/j jytj supx�2yt

jx˛��˛ j � C j�.y/j jytj ! 0;

as t ! 0. The last term can be estimated by C j�.y/j jyj, which is again integrable.Lebesgue’s dominated convergence theorem implies that

limt!0

supx2R

jx˛. f � �t.x/� f .x//j D 0:

The same can be shown for the derivatives, which proves the statement in (a).For (b) let us apply the Lp-norm of both sides in (1.5.3) and Minkowski’s

inequality to obtain

k f � �t.x/ � f .x/kp D

Z

R

�.y/ . f .x � yt/ � f .x// dy

p

�Z

R

j�.y/j k f .x � yt/� f .x/kp dy: (1.5.4)

We will use that

limh!0

kThf � f kp D 0 (1.5.5)

for all f 2 Lp.R/ with 1 � p < 1. Equation (1.5.5) holds if f is continuous withcompact support. Indeed, in this case f is uniformly continuous and thus for all� > 0 there exists ı > 0 such that for all jhj < ı and all t 2 R, j f .t � h/� f .t/j < �.We may suppose that ı < 1. If the support of f is contained in Œa; b�, then supp j f .��h/� f .�/j � Œa � 1; b C 1� and

�Z

R

j f .t � h/� f .t/jp dt

�1=p

� �

�Z bC1

a�11 dt

�1=p

D � .b C 2 � a/1=p ;

whenever jhj < ı. Thus (1.5.5) is true for continuous functions with compactsupport. Since these functions are dense in Lp.R/, for all � > 0 there exists acontinuous function g with compact support such that

k f � gkp < �:

Then by the triangle inequality,

kThf � f kp � kThf � Thgkp C kThg � gkp C kg � f kp

� 2k f � gkp C kThg � gkp

< 3�

1.5 Inequalities with Respect to Hardy Spaces 29

if h is small enough, which shows (1.5.5). Hence

limt!0

j�.y/j k f .x � yt/ � f .x/kp D 0:

Since

j�.y/j k f .x � yt/ � f .x/kp � 3 j�.y/j k f kp ;

(b) follows from (1.5.4) and from Lebesgue’s dominated convergence theorem.Applying Definition 1.4.6 and (a) for M�t, we conclude

limt!0

f � �t.h/ D limt!0

f . M�t � h/ D f .h/

for all h 2 S.R/, which shows (c). �For 1 < p � 1 the Hardy spaces coincide with the Lp.R/ spaces.

Theorem 1.5.3 If 1 < p < 1, then Hp.R/ � Lp.R/ and

k f kp � k f kHp� Cp k f kp :

Proof The right-hand side follows from Theorem 1.5.1. For the left-hand side let� 2 S.R/ with

R

R� d� D 1 and f 2 Hp.R/. We may suppose that k f kHp � 1. By

the definition of the Hardy space the set ff � �t W t > 0g lies in the closed unit ballof Lp.R/. Since Lp.R/ is the dual space of Lp0.R/ .1=p C 1=p0 D 1/, the Banach-Alaoglu theorem says that the closed unit ball of Lp.R/ is compact in the weak�topology (see e.g. Rudin [276]). This implies that there exists a sequence .tj; j 2 N/,tj ! 0 such that f � �tj converges to some f0 2 Lp.R/ in the weak� topology. Thus

limtj!0

Z

R

f � �tj.x/g.x/ dx DZ

R

f0.x/g.x/ dx

for all g 2 Lp0.R/ S.R/. On the other hand, by Theorem 1.5.2,

limt!0

f � �t D f in S0.R/:

Hence f D f0 2 Lp.R/. Again by Theorem 1.5.2,

limt!0

f � �t D f in the Lp-norm:

Hence

k f kp � supt>0

k f � �tkp �

supt>0

j f � �tj

p

D k f kHp; (1.5.6)

which completes the proof of the theorem. �

The theorem is not true for p D 1, H1.R/ does not coincide with L1.R/.Theorem 1.6.10 implies that the elements of H1.R/ have integral 0, thus in viewof the next theorem, H1.R/ is a proper subspace of L1.R/.

Theorem 1.5.4 For p D 1, H1.R/ � L1.R/ and

k f k1 � k f kH1 :

Proof It is known that the dual space of L1.R/ is the space of finite Borel measures.As in the preceding theorem we get that f 2 H1.R/ coincides with a measure and

limtj!0

Z

R

f � �tj.x/g.x/ dx DZ

R

g.x/ d .x/ (1.5.7)

for all g 2 L1.R/. It is enough to prove that is absolutely continuous with respectto the Lebesgue measure. Since the measure

F 7!Z

Fsupt>0

j f � �t.x/j dx

is absolutely continuous with respect to the Lebesgue measure as well, for all � > 0there exists a ı > 0 such that for all measurable sets F with �.F/ < ı, we haveR

F supt>0 j f � �t.x/j dx < �. If �.E/ D 0, then there exists an open set U such thatE � U and �.U/ < ı. For any bounded function g supported in U, (1.5.7) implies

ˇ

ˇ

ˇ

ˇ

Z

R

g.x/ d .x/

ˇ

ˇ

ˇ

ˇ

� kgk1Z

Usupt>0

j f � �t.x/j dx � � kgk1 :

Then

j .U/j D sup

�ˇ

ˇ

ˇ

ˇ

Z

R

g.x/ d .x/

ˇ

ˇ

ˇ

ˇ

W g 2 C.R/; supp g � U; kgk1 � 1

�

< �;

where j j denotes the total variation of . Since � is arbitrary, j .E/j D 0, thus is absolutely continuous with respect to �. The theorem follows as in (1.5.6). �

1.6 Atomic Decomposition

The atomic decomposition is a useful characterization of the Hardy spaces bythe help of which some boundedness results, duality theorems, inequalities andinterpolation results can be proved. The atoms are relatively simple and easy tohandle functions. If we have an atomic decomposition, then we have to prove severaltheorems for atoms, only.

1.6 Atomic Decomposition 31

For the atomic decomposition of the Hardy space Hp.R/, we will need thefollowing two covering lemmas.

Lemma 1.6.1 (Vitali-Wiener Covering Lemma) Let � � R be an open setwith finite measure. If I.x; r.x// � � .x 2 �/, then one can choose a series�

I.xi; r.xi//; i 2 N�

such that the intervals I.xi; r.xi// are disjoint and

� �[

i2NI.xi; 4r.xi//:

Proof Since supx2� r.x/ < 1, there exists x0 2 � such that

r.x0/ >1

2supx2�

r.x/:

We define two series .�i � �/ and .xi 2 �i/ in the following way. Let �0 D �

and suppose that we defined�j and xj 2 �j for j D 1; : : : ; i � 1. Let

�i WD � ni�1[

jD0I.xj; 4r.xj// (1.6.1)

and xi 2 �i such that

r.xi/ >1

2supx2�i

r.x/:

If�i D ; for some i, then the lemma is proved. Otherwise observe that j < i impliesr.xi/ < 2r.xj/. Suppose that

I.xi; r.xi//\ I.xj; r.xj// ¤ ;; .j < i/:

In this case

jxj � xij � r.xj/C r.xi/ � 3r.xj/;

which implies that xi 2 I.xj; 4r.xj//. However, this is a contradiction with xi 2 �i

and (1.6.1). Hence the intervals .I.xi; r.xi//; i 2 N/ are really disjoint. From this itfollows that r.xi/ ! 0 .i ! 1/.

We will show that

� n1[

jD0I.xi; 4r.xi// D ;:

If this is not true then there exists y 2 �n[1jD0I.xi; 4r.xi//. If i is large enough, then

r.y/ > 2r.xi/, that is a contradiction with

r.xi/ >1

2supx2�i

r.x/ >1

2r.y/:

The proof of Lemma 1.6.1 is complete. �Let us denote the complement of a set H by Hc.

Lemma 1.6.2 (Whitney Covering Lemma) Let � � R be an open set with finitemeasure. Then there exist two series .xi 2 �/ and .ri > 0/ such that

(i) the intervals I.xi; ri=4/ are disjoint and � D [i2NI.xi; ri/,(ii) I.xi; 18ri/\�c D ; and I.xi; 54ri/ \�c ¤ ;,

(iii) there exists a constant M such thatX

i

1fI.xi;18ri/g.x/ � M .x 2 �/:

Proof For x 2 � let

%.x; �c/ WD inffjx � yj W y 2 �cgbe the distance between x and �c and let

S.x/ WD %.x; �c/=144 > 0:

By Lemma 1.6.1, from the intervals I.x; S.x// .x 2 �/ one can choose a series.I.xi; S.xi/// of disjoint intervals such that [iI.xi; 4S.xi// �. Since 4S.xi/ D%.xi; �

c/=36, we have I.xi; 4S.xi// � � and so

[

i

I.xi; 4S.xi// D �:

Thus (i) holds with ri WD 4S.xi/. Since 18ri D %.xi; �c/=2 and 54ri D 3=2%.xi; �

c/

we have (ii).To prove (iii), let x 2 I.xi; 18ri/. In this case

36ri D %.xi; �c/ � jxi � xj C %.x; �c/ � 18ri C %.x; �c/;

hence 18ri � %.x; �c/. If x; y 2 I.xi; 18ri/, then jx � yj � 2%.x; �c/ and

I.xi; 18ri/ � I.x; 2%.x; �c//:

If x belongs to M intervals I.xi; 18ri/, then these intervals are contained inI.x; 2%.x; �c//. On the other hand,

%.x; �c/ � jx � xij C %.xi; �c/ < 54ri

implies ri=4 > %.x; �c/=216. Note that the intervals I.xi; ri=4/ are disjoint.Computing the length of fI.xi; ri=4/ W x 2 I.xi; 18ri/g we can see that

2M%.x; �c/=216 �X

i

ri=2 � 4%.x; �c/:

Hence M � 432 which finishes the proof of Lemma 1.6.2. �Note that (iii) is called the bounded overlapping property.Assume that f 2 Hp.R/ and let

� WD f f� > �g:

We can choose two series .xi; i 2 N/ and .ri; i 2 N/ such that Lemma 1.6.2 holds.Let � 2 C1

c .R/ (i.e. � is infinitely differentiable with compact support) such thatsupp � � I.0; 2/, 0 � � � 1 and �.x/ D 1 if jxj < 1. If

�i.x/ WD �

�

x � xi

ri

�

.i 2 N/;

then �i.x/ D 1 in case x 2 I.xi; ri/, supp �i � I.xi; 2ri/ and

1 �X

i

�i.x/ � M; .x 2 �/:

For

�i.x/ WD(

�i.x/=P

j �j.x/; if x 2 �I0; if x 2 �c;

we obtain that �i 2 C1c .R/, supp �i � I.xi; 2ri/, 0 � �i � 1,

P

i �i D 1� and1=M � �i.x/ � 1 if x 2 I.xi; ri/.

Let us denote the set of N.p/-order polynomials by PN.p/. For a fixed j 2 N, weintroduce the following weighted norm

kPk.j/ WD R

RjPj2�j d�R

R�j d�

!1=2

.P 2 PN.p//; (1.6.2)

and let .�j;l; l D 1; : : : ;N.p// be an orthonormal basis, i.e.

R

R�j;l�j;k�j d�R

R�j d�

D�

1; if k D lI0; if k ¤ l:


Let

Pj.x/ WDN.p/X

lD1

�R

Rf�j;l�j d�R

R�j d�

�

�j;l.x/ (1.6.3)

and

Pi;j.x/ DN.p/X

lD1

�R

R. f � Pj/�j;l�i�j d�

R

R�j d�

�

�j;l.x/; (1.6.4)

where �i 2 C1c .R/ will be chosen later. For these polynomials we need the

following lemmas.

Lemma 1.6.3 For all polynomials Q 2 PN.p/,

Z

R

. f � Pj/Q�j d� D 0 (1.6.5)

andZ

R

. f � Pj/Q�i�j d� DZ

R

Pi;jQ�j d�: (1.6.6)

Proof It is enough to check the equalities for Q D �j;k, k D 1; : : : ;N.p/:

Z

R

Pj�j;k�j d� DZ

R

N.p/X

lD1

�R

Rf�j;l�j d�R

R�j d�

�

�j;l�j;k�j d�

D�R

Rf�j;k�j d�R

R�j d�

�Z

R

�j;k�j;k�j d�

DZ

R

f�j;k�j d�:

For the other equality we have

Z

R

Pi;j�j;k�j d� DZ

R

N.p/X

lD1

�R


R

R�j d�

�

�j;l�j;k�j d�

D�R

R. f � Pj/�j;k�i�j d�

R

R�j d�

�Z

R

�j;k�j;k�j d�

DZ

R

. f � Pj/�j;k�i�j d�;

which completes the proof. �

One can show that Pj and Pi;j are the unique polynomials that satisfy (1.6.5)and (1.6.6), respectively.

Lemma 1.6.4 There exists a constant C independent of l D 1; : : : ;N.p/, j 2 N,� > 0, such that

supx2R

r˛j

ˇ

ˇ

ˇ�.˛/j .x/

ˇ

ˇ

ˇ � C .˛ 2 N/ ; (1.6.7)

sup˛�m;y2I.xj;2rj/

r˛j

ˇ

ˇ

ˇ�.˛/j;l .y/

ˇ

ˇ

ˇ � C .m 2 N/ (1.6.8)

and

sup˛�m;y2R

r˛j

ˇ

ˇ

ˇ�.˛/j;l .y/�j.y/

ˇ

ˇ

ˇ � C .m 2 N/ : (1.6.9)

Proof Inequality (1.6.7) follows from the definition of �j and from the fact � 2C1

c .R/. It follows from (1.6.2) that

1 D

�j;l

2

.j/� M�1

jI.xj; 2rj/jZ

I.xj;rj/

ˇ

ˇ�j;l.y/ˇ

ˇ

2dy

D CZ

I.0;1/

ˇ

ˇ�j;l.xj C rjt/ˇ

ˇ

2dt:

Since on a finite dimensional space every norms are equivalent, so the norms

�Z

I.0;1/jP.t/j2 dt

�1=2

and supjtj<2

jP.t/j

are equivalent on PN.p/. Thus there is a constant C independent of P such that

supjtj<2

jP.t/j � C

�Z

I.0;1/jP.t/j2 dt

�1=2

.P 2 PN.p//:

By Bernstein’s inequality,

sup˛�m;jtj<2

ˇ

ˇP.˛/.t/ˇ

ˇ � C supjtj<2

jP.t/j .P 2 PN.p//:

Using these two inequalities and taking P.t/ D �j;l.xj C rjt/, we have proved

sup˛�m;y2I.xj;2rj/

r˛j

ˇ

ˇ

ˇ�.˛/j;l .y/

ˇ

ˇ

ˇ D sup˛�m;jtj<2

ˇ

ˇP.˛/.t/ˇ

ˇ

� C supjtj<2

jP.t/j

� C

�Z

I.0;1/jP.t/j2 dt

�1=2

� C;

which is (1.6.8). Since supp �j � I.xj; 2rj/, (1.6.8) implies (1.6.9). �

Lemma 1.6.5 There exists a constant C independent of j; � such that

supy2I.xj;2rj/

jPj.y/j � C�: (1.6.10)

Proof It is enough to show that

ˇ

ˇ

ˇ

ˇ

R

Rf�j;l�j d�R

R�j d�

ˇ

ˇ

ˇ

ˇ

� C�

for all l D 1; : : : ;N.p/, because of (1.6.3) and (1.6.8). For a fixed z 2 I.xj; 54rj/\�c

the left-hand side of the preceding inequality can be written as

ˇ

ˇ

ˇ

ˇ

rjR

R�j d�

Z

R

f .z � rjx/�j;l.z � rjx/�j.z � rjx/ dx

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

Z

R

f .z � rjx/�j;l.x/ dx

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ f � � j;lrj.z/ˇ

ˇ

ˇ ;

where

� j;l.x/ WD rjR

R�j d�

�j;l.z � rjx/�j.z � rjx/:

Since

rjR

R�j d�

� MrjR

I.xj;rj/1 d�

� C

and � j;l 2 C1c .R/, by (1.6.7) and (1.6.9) we have

� j;l

Km� C;

where C is independent of z; j; �. Hence

ˇ

ˇ

ˇ

ˇ

R

Rf�j;l�j d�R

R�j d�

ˇ

ˇ

ˇ

ˇ

� Cf�.z/ � C�

and (1.6.10) is shown. �Starting with a set �0, we repeat the definition of �j. Let �0 � be an open set

with finite measure. In the same way as before, for this�0 we can choose two series.yi; i 2 N/ and .si; i 2 N/ such that Lemma 1.6.2 holds. Let

#i.x/ WD �

�

x � yi

si

�

.i 2 N/

and

�i.x/ WD(

#i.x/=P

j #j.x/; if x 2 �0I0; if x 2 .�0/c:

It is easy to see that the functions �i satisfy the same properties as �i. Amongst others�i 2 C1

c .R/ and the analogue of (1.6.7) reads as follows:

supx2R

s˛i

ˇ

ˇ

ˇ�.˛/i .x/

ˇ

ˇ

ˇ � C .˛ 2 N/ : (1.6.11)

In what follows, we use these functions �i in the definition of Pi;j in (1.6.4).

Lemma 1.6.6 If I.xj; 2rj/\ I.yi; 2si/ ¤ ;, then I.xj; 2rj/ � I.yi; 18si/ and rj � 4si.

Proof If I.xj; 2rj/\ I.yi; 2si/ ¤ ;, then

jxj � yij < 2si C 2rj:

Since the series .xj/ and .si/ satisfy (ii) of Lemma 1.6.2 and� � �0, we have

%.xj; .�0/c/ � %.xj; �

c/ � 18rj:

Thus

18rj � %.xj; .�0/c/ � jxj � yij C %.yi; .�

0/c/ � 2rj C 56si

and so rj � 4si. If y 2 I.xj; 2rj/, then

jy � yij � ˇ

ˇy � xj

ˇ

ˇC ˇ

ˇxj � yi

ˇ

ˇ � 4rj C 2si � 18si

and the lemma is proved. �


Lemma 1.6.7 There exists a constant C independent of i; j; � such that

supy2R

ˇ

ˇPi;j.y/�j.y/ˇ

ˇ � C�: (1.6.12)

Proof Taking into account the fact supp �j � I.xj; 2rj/, (1.6.4) and (1.6.8) we cansee that the lemma follows from

ˇ

ˇ

ˇ

ˇ

R


R

R�j d�

ˇ

ˇ

ˇ

ˇ

� C�

for all l D 1; : : : ;N.p/. In order to show this inequality, by (1.6.9) and Lemma 1.6.4it is enough to show that

ˇ

ˇ

ˇ

ˇ

R

Rf�j;l�i�j d�R

R�j d�

ˇ

ˇ

ˇ

ˇ

� C�:

Similar to the above method, for z 2 I.xj; 54rj/ \ �c the left-hand side of thepreceding inequality can be written as

ˇ

ˇ

ˇ

ˇ

rjR

R�j d�

Z

R

f .z � rjx/�j;l.z � rjx/�i.z � rjx/�j.z � rjx/ dx

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

Z

R

f .z � rjx/ j;l.x/ dx

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ f � j;lrj.z/ˇ

ˇ

ˇ ;

where

j;l.x/ WD rjR

R�j d�

�j;l.z � rjx/�i.z � rjx/�j.z � rjx/:

We have proved in Lemma 1.6.6 that rj � 4si. Using this and (1.6.11), we can showas above that

j;l

Km� C;

where C is independent of z; i; j; �. Hence

ˇ

ˇ

ˇ

ˇ

R

Rf�j;l�i�j d�R

R�j d�

ˇ

ˇ

ˇ

ˇ

� Cf�.z/ � C�

and (1.6.12) is verified. �


Lemma 1.6.8 We have

X

i

X

j

Pi;j�j D 0 a.e. and in S0.R/: (1.6.13)

Proof Since the intervals I.xj; rj/ have the bounded overlapping property andsupp �j � I.xj; 2rj/, we can see that for any fixed x 2 R,

ˇ

ˇfj W �j.x/ ¤ 0gˇˇ � M:

We can suppose that I.xj; 2rj/ \ I.yi; 2si/ ¤ ;, since otherwise Pi;j;D 0. ByLemma 1.6.6 and by the bounded overlapping property, for every fixed j,

ˇ

ˇfi W I.xj; 2rj/\ I.yi; 2si/ ¤ ;gˇˇ � M:

Then,

X

i

X

j

Pi;j.x/�j.x/ DX

j

X

i

Pi;j.x/

!

�j.x/:

It is enough to show that

X

i

Pi;j.x/ D 0 (1.6.14)

for all x 2 R. It follows from (1.6.6) that

Z

R

. f � Pj/Q

X

i

�i

!

�j d� DZ

R

X

i

Pi;j

!

Q�j d�

for all polynomials Q 2 PN.p/. Recall thatP

i �i D 1�0 and supp �j � � � �0. Weget by (1.6.5) that

Z

R

X

i

Pi;j

!

Q�j d� DZ

�0

. f � Pj/Q�j d� DZ

R

. f � Pj/Q�j d� D 0

for all Q 2 PN.p/. Equality (1.6.14) follows easily from this. Moreover, by (1.6.12),

Z

R

X

i

X

j

ˇ

ˇPi;j�j

ˇ

ˇ d� � CM2�j�j

and we can conclude that the equality in (1.6.13) holds in L1.R/ as well as in thesense of distributions. This completes the proof of Lemma 1.6.8. �

Now we are ready to prove the atomic decomposition of the Hardy space Hp.R/.

Definition 1.6.9 A bounded measurable function a is a p-atom .0 < p < 1/ ifthere exists an interval I � R such that

(i) supp a � I,(ii) kak1 � jIj�1=p,

(iii)R

I a.x/xk dx D 0 for all natural numbers k � N.p/.

Recall that N.p/ D b1=p � 1c.

Theorem 1.6.10 A tempered distribution f 2 S0.R/ is in Hp.R/ .0 < p � 1/ if andonly if there exist a sequence .ak; k 2 N/ of p-atoms and a sequence . k; k 2 N/ ofreal numbers such that

1X

kD0j kjp < 1 and

1X

kD0 kak D f in S0.R/: (1.6.15)

Moreover,

k f kHp� inf

1X

kD0j kjp

!1=p

; (1.6.16)

where the infimum is taken over all decompositions of f of the form (1.6.15).

Proof First we verify that

�Ca

p� Cp (1.6.17)

for all p-atom a. We can suppose that the support of a, the interval I is centred at theorigin. For � 2 S.R/ with k�kKm � 1 and

R

R� d� ¤ 0, we have by the definition

of the atom thatZ

2I

ˇ

ˇ�Caˇ

ˇ

pd� � CpjIj�1j2Ij � Cp: (1.6.18)

Recall that rI.x; h/ D I.x; rh/. We may suppose that m D N.p/C 1. Let Pm�1.z/ bethe .m �1/ order Taylor polynomial of �.w � z/ in z. Using the definition of Km andTaylor’s formula, we have for z 2 I.0; �/ and w 2 .I.0; 2�//c that

j�.w � z/ � Pm�1.z/j Dˇ

ˇ�.˛/.w � �z/ˇ

ˇ

mŠjzjm � Cjwj�m�1jzjm;

where � 2 .0; 1/. Applying this inequality and (iii) of the definition of the atom, weget for x … 2I that

j.a � �t/.x/j Dˇ

ˇ

ˇ

ˇ

1

t

Z

R

a.y/�x � y

t

�

dy

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

1

t

Z

R

a.y/

�x � y

t

�

� Pm�1y

t

��

dy

ˇ

ˇ

ˇ

ˇ

� C1

t

ˇ

ˇ

ˇ

x

t

ˇ

ˇ

ˇ

�m�1 Z

Ija.y/j

ˇ

ˇ

ˇ

y

t

ˇ

ˇ

ˇ

mdy (1.6.19)

� Cjxj�m�1jIj�1=pCmC1:

HenceZ

.2I/c

ˇ

ˇ�Ca.x/ˇ

ˇ

pdx � Cp

Z

.2I/cjxj�mp�pjIj�1CmpCp dx � Cp;

because m D N.p/C 1, which proves (1.6.17).Suppose that f 2 S0.R/ has a decomposition satisfying (1.6.15). By (1.6.17) it is

easy to see that

f � �t D1X

kD0 kak � �t

and

�Cf �1X

kD0j kj�Cak:

Since 0 < p � 1,

Z

R

ˇ

ˇ�Cfˇ

ˇ

pd� �

1X

kD0j kjp

Z

R

ˇ

ˇ�Cakˇ

ˇ

pd� � Cp

1X

kD0j kjp :

Thus f 2 Hp.R/ and one side of (1.6.16) holds.On the other hand, assume that f 2 Hp.R/ \ L2.R/. Recall that Hp.R/ \ L2.R/

is dense in Hp.R/. We will apply the lemmas presented above for

� D �kC1 WD ˚

f� > 2kC1� and �0 D �k WD ˚

f� > 2k�

.k 2 Z/. The points xj, yi will be denoted by xkC1j , xk

i , the radiuses rj, si by rkC1j ,

rki , the functions �j, �i by �kC1

j , �ki and the polynomials Pj, Pi;j by PkC1

j , PkC1i;j ,

respectively.


Set

gk WD f1�k �X

i

Pki �

ki D

X

i

. f � Pki /�

ki

and

bk WD f1�ck

CX

i

Pki �

ki :

In this case f D gk C bk. By the bounded overlapping property and by Lemma 1.6.5we conclude

jbkj � C2k:

Hence

limk!�1 gk.x/ D f .x/ a.e.

Since supp gk � �k and

j�kj � k f�kpp

2kp! 0 as k ! 1;

we obtain

limk!1 gk.x/ D 0 a.e.

Consequently,

f .x/ D1X

kD�1.gk.x/ � gkC1.x// a.e. (1.6.20)

Recall thatP

i �ki D 1�k and supp �kC1

j � �kC1 � �k. Using Lemma 1.6.8, weconclude

gk � gkC1 DX

i

. f � Pki /�

ki �

X

j

. f � PkC1j /�kC1

j

DX

i

0

@. f � Pki /�

ki �

X

j

. f � PkC1j /�k

i �kC1j

1

A


DX

i

0

@. f � Pki /�

ki �

X

j

. f � PkC1j /�k

i � PkC1i;j

�

�kC1j

1

A

DWX

i

hki ;

where

hki WD . f � Pk

i /�ki �

X

j

. f � PkC1j /�k

i � PkC1i;j

�

�kC1j :

Recall that supp �ki � I.xk

i ; 2rki /. If I.xkC1

j ; 2rkC1j / \ I.xk

i ; 2rki / D ;, then PkC1

i;j D0 by (1.6.4). Lemma 1.6.6 says that if I.xkC1

j ; 2rkC1j / \ I.xk

i ; 2rki / D ;, then

I.xkC1j ; 2rkC1

j / � I.xki ; 18rk

i / D ;. Thus supp PkC1i;j �kC1

j � I.xki ; 18rk

i / and

supp hki � I.xk

i ; 18rki /:

The functions hki can be written as

hki D f �k

i 1�ckC1

� Pki �

ki C �k

i

X

j

PkC1j �kC1

j CX

j

PkC1i;j �kC1

j :

Then by Theorem 1.5.2,

ˇ

ˇ

ˇ f �ki 1�c

kC1

ˇ

ˇ

ˇ � Cf�1�ckC1

� C2k:

By the bounded overlapping property of the balls I.xkC1j ; rkC1

j / and (1.6.10)and (1.6.12) we conclude that

ˇ

ˇhki

ˇ

ˇ � C2k1I.xki ;18rk

i /� Cf�: (1.6.21)

Therefore

f .x/ D1X

kD�1

X

i

hki .x/ a.e. (1.6.22)

It follows from (1.6.5) and (1.6.6) that

Z

R

hki .x/Q.x/ dx D 0

for all Q 2 PN.p/. Taking

ki WD C02

kˇ

ˇI.xki ; 18rk

i /ˇ

ˇ

1=pand ak

i WD hki

ki

;

we have by (1.6.22) that

f .x/ D1X

kD�1

X

i

ki ak

i .x/ a.e.; (1.6.23)

where each aki is a p-atom. By Abel rearrangement,

1X

kD�1

X

i

ˇ

ˇ ki

ˇ

ˇ

p D Cp

1X

kD�1

X

i

2kpˇ

ˇI.xki ; r

ki /ˇ

ˇ (1.6.24)

D Cp

1X

kD�12kpˇ

ˇf f� > 2kgˇˇ

D Cp

1X

kD�1.2p/k

ˇ

ˇf.2p/k < f p� � .2p/kC1gˇˇ

� Cp k f�kpp :

We have to prove yet that (1.6.23) holds in the sense of tempered distributions.The inequalities

ˇ

ˇPki .x/�

ki .x/

ˇ

ˇ � C2k�ki .x/ � Cf�.x/�

ki .x/

andˇ

ˇ

ˇ

ˇ

ˇ

X

i

Pki .x/�

ki .x/

ˇ

ˇ

ˇ

ˇ

ˇ

� Cf�.x/;

imply that bk.x/ � Cf�.x/. From this it follows that

ˇ

ˇgk.x/ˇ

ˇ D ˇ

ˇ f .x/ � bk.x/ˇ

ˇ � Cf�.x/:

Note that f 2 Hp.R/\ L2.R/ implies f� 2 L1.R/. It follows from (1.6.21) and fromLebesgue dominated convergence theorem that (1.6.20) and (1.6.23) holds in L1.R/as well as in the sense of tempered distributions.

Suppose now that f 2 Hp.R/ is arbitrary. Since L2.R/\Hp.R/ is dense in Hp.R/

by Theorem 1.4.13, there exist fk 2 Hp.R/\ L2.R/, such that

k f � fkkHp! 0 as k ! 1:

We may assume that

k f0kpHp

� 3

2k f kp

Hp

and

k fkC1 � fkkpHp

� 2�k�1 k f kpHp

.k 2 N/:

Set g0 WD f0 and gk WD fk � fk�1 .k � 1/. Then

f D1X

kD0gk in the Hp-norm:

Since gk 2 Hp.R/\ L2.R/, it has an atomic decomposition of the form

gk D1X

jD0 k

j akj in S0.R/;

where akj are p-atoms and

1X

jD0

ˇ

ˇ kj

ˇ

ˇ

p � Cp kgkkpHp:

Hence, by Theorem 1.4.12, f can be written as

f D1X

kD0

1X

jD0 k

j akj in S0.R/

and

1X

kD0

1X

jD0

ˇ

ˇ kj

ˇ

ˇ

p � Cp

1X

kD0kgkkp

Hp� Cp k f kp

Hp:

This finishes the proof of Theorem 1.6.10. �

Remark 1.6.11 The convergence in (1.6.23) holds in the L2.R/-norm, too. Simi-larly, if f 2 Hp.R/ \ Lq.R/ .1 < q < 1/, then it holds in the Lq.R/-norm, iff 2 Hp.R/ \ H1.R/, then in the L1.R/-norm. The atomic decomposition (1.6.15)converges also in the Hp.R/-norm when 0 < p � 1.

This proof of the atomic decomposition of the Hp.R/ space is due to Latter[210] (see also Lu [233] or Weisz [355]). For other forms or proofs of thisatomic decomposition we refer to Coifman and Weiss [67], Coifman [66], Wilson

[382, 383], Stein [309] and for martingales to Weisz [347]. We could suppose thatthe integral in (iii) of Definition 1.6.9 is zero for all natural numbers k � N, whereN � N.p/. The best possible choice of such numbers N is N.p/.

Changing (ii) in Definition 1.6.9, we obtain the definition of .p; q/-atoms.

Definition 1.6.12 A bounded measurable function a is a .p; q/-atom .0 1/ if there exists an interval I � R such that

(i) supp a � I,(ii) kakq � jIj1=q�1=p,

(iii)R

I a.x/xk dx D 0 for all natural numbers k � N.p/.

If q D 1, then we get back the definition of p-atoms. It is easy to see that everyp-atom is also a .p; q/-atom. Indeed,

�Z

Ijajq d�

�1=q

� jIj.�q=pC1/=q D jIj�1=pC1=q ;

where I is the support of a. The atomic decomposition theorem holds for theseatoms, too.

Theorem 1.6.13 A tempered distribution f 2 S0.R/ is in Hp.R/ .0 < p � 1/ if andonly if there exist a sequence .ak; k 2 N/ of .p; q/-atoms and a sequence . k; k 2 N/

of real numbers such that

1X

kD0j kjp < 1 and

1X

kD0 kak D f in S0.R/:

Moreover, if 0 < p � 1, then

k f kHp� inf

1X

kD0j kjp

!1=p

;

where the infimum is taken over all atomic decompositions of f .

Proof The proof is essentially the same as that of Theorem 1.6.10. The twodifferences are the following. In (1.6.18) we use Hölder’s inequality to obtain

Z

2I

ˇ

ˇ�Caˇ

ˇ

pd� �

�Z

2I

ˇ

ˇ�Caˇ

ˇ

qd�

�p=q

j2Ij1�p=q

� Cp;q

�Z

2Ijajq d�

�p=q

jIj1�p=q � Cp;q:

1.7 Interpolation Between Hardy Spaces 47

Instead of (1.6.19) we observe that

j.a � �t/.x/j � C jxj�m�1Z

Ija.y/j jyjm dy

� C jxj�m�1�Z

Ija.y/jq dy

�1=q �Z

Ijyjmq0

dy

�1=q0

� Cjxj�m�1jIj1=q�1=pCmC1=q0

;

where 1=qC1=q0 D 1. Since every p-atom is a .p; q/-atom, the proof can be finishedas in Theorem 1.6.10. �

1.7 Interpolation Between Hardy Spaces

In this section and later in Sect. 3.5 we will need the concept of Lorentz spacesLp;q.R/. The Lp;1.R/ space is already defined in (1.1.1). For a moment, let us denotethat space by L�

p .R/.

Definition 1.7.1 For a measurable function f the non-increasing rearrangement isdefined by

Qf .t/ WD inf f� W jfj f j > �gj � tg :It is easy to see that Qf is non-increasing, continuous on the right and it is

equimeasurable with f , namely,ˇ

ˇf Qf > �gˇˇ D jfj f j > �gj .� � 0/: (1.7.1)

Indeed, by the definition of Qf , we have Qf .jfj f j > �gj/ � � and thus jf Qf > �gj �jfj f j > �gj. On the other hand, since Qf is continuous on the right, Qf .jf Qf > �gj/ � �

and so jfj f j > �gj � jf Qf > �gj.Note that if Qf is continuous at a point t then � D Qf .t/ is equivalent to t D jfj f j >

�gj.Definition 1.7.2 The Lorentz space Lp;q.R/ consists of all measurable functions ffor which

k f kp;q WD�Z 1

0

Qf .t/qtq=p dt

t

�1=q

< 1 if 0 0

t1=p Qf .t/ < 1 if 0 < p < 1:

In case of p D 1, let Lp;1.R/ WD L1.R/.

Proposition 1.7.3 One has

Lp;p.R/ D Lp.R/; Lp;1.R/ D L�p .R/ .0 �gj � t. Thus

� jfj f j > �gj1=p D �ˇ

ˇf Qf > �gˇˇ1=p � t1=p Qf .t/

and so k f kL�

p� k f kp;1. On the other hand, for a given � > 0, we can choose t such

that Qf is continuous in t and k f kp;1 � t1=p Qf .t/C�. Set � D Qf .t/. Then jf Qf > �gj D tand

k f kp;1 � t1=p Qf .t/C � D �ˇ

ˇf Qf > �gˇˇ1=p C � � k f kL�

pC �

which proves the second equality. �One can show (see e.g. Bennett and Sharpley [24] or Weisz [347]) that the

Lorentz spaces Lp;q.R/ increase as the second exponent q increases, namely, for0 < p < 1 and 0 < q1 � q2 � 1 one has Lp;q1 .R/ � Lp;q2 .R/.

Now we generalize the definition of Hardy spaces.

Definition 1.7.4 For 0 < p; q � 1 the Hardy-Lorentz spaces Hp;q.R/ consist ofall tempered distributions for which

k f kHp;qWD k f�kp;q < 1:

Of course, for p D q we get back the Hardy spaces Hp.R/. In this sectionthe interpolation spaces between the Hardy-Lorentz spaces are identified. Thebasic definitions and theorems of interpolation theory are given without proofs asfollows. For the details see Bennett and Sharpley [24] and Bergh and Löfström[29].

Suppose that A0 and A1 are quasi-normed spaces embedded continuously in atopological vector space A. In the real method of interpolation, the interpolationspaces between A0 and A1 are defined by means of an interpolating functionK.t; f ;A0;A1/.

Definition 1.7.5 For f 2 A0 C A1, the interpolating function is defined by

K.t; f ;A0;A1/ WD inff Df0Cf1

fk f0kA0 C tk f1kA1g ;

where the infimum is taken over all choices of f0 and f1 such that f0 2 A0, f1 2 A1and f D f0 C f1.

Definition 1.7.6 The interpolation space .A0;A1/�;q is introduced as the space ofall functions f in A0 C A1 such that

k f k.A0;A1/�;q WD�Z 1

0

t��K.t; f ;A0;A1/�q dt

t

�1=q

< 1

if 0 < q < 1 and

k f k.A0;A1/�;1 WD supt>0

t��K.t; f ;A0;A1/ < 1;

where 0 < � < 1. We use the conventions .A0;A1/0;q D A0 and .A0;A1/1;q D A1for each 0 < q � 1.

Suppose that B0 and B1 are also quasi-normed spaces embedded continuously ina topological vector space B.

Definition 1.7.7 A map

T W A0 C A1 �! B0 C B1

is said to be quasilinear from .A0;A1/ to .B0;B1/ if for given a 2 A0CA1 and ai 2 Ai

with a0 C a1 D a there exist bi 2 Bi satisfying

Ta D b0 C b1

and

kbikBi� Ki kaikAi

.Ki > 0; i D 0; 1/:

Clearly, if T is linear and bounded from Ai to Bi .i D 0; 1/, then T is quasilinear.The following theorem shows that the boundedness of a quasilinear operator ishereditary for the interpolation spaces. Under some conditions this result will beapplied for sublinear operators. An operator T is called sublinear if

jT.a1 C a2/j � jT.a1/j C jT.a2/j :

Theorem 1.7.8 If 0 < q � 1, 0 � � � 1 and T is a quasilinear map from .A0;A1/to .B0;B1/, then

T W .A0;A1/�;q �! .B0;B1/�;q

and

kTak.B0;B1/�;q � K1��0 K�

1 kak.A0;A1/�;q :

The reiteration theorem below is one of the most important general results in theinterpolation theory. It says that the interpolation space of two interpolation spacesis also an interpolation space of the original spaces.

Theorem 1.7.9 (Reiteration Theorem) Suppose that 0 � �0 < �1 � 1, 0 <q0; q1 � 1 and Xi D .A0;A1/�i;qi .i D 0; 1/. If 0 < � < 1 and 0 < q � 1, then

.X0;X1/�;q D .A0;A1/�;q

where

� D .1 � �/�0 C ��1:

Theorem 1.7.10 (Wolff) Let A1, A2, A3 and A4 be quasi-Banach spaces satisfyingA1 \ A4 � A2 \ A3. Suppose that

A2 D .A1;A3/�;q; A3 D .A2;A4/ ;r

for any 0 < �; < 1 and 0 < q; r � 1. Then

A2 D .A1;A4/�;q; A3 D .A1;A4/�;r

where

� D �

1 � � C � ; � D

1 � � C � :

The proofs of these theorems can be found e.g. in Bennett and Sharpley [24] andin Bergh and Löfström [29]. In the sequel the following two Hardy type inequalitiesare needed.

Lemma 1.7.11 (Hardy’s Inequality) If 1 � q < 1, r > 0 and f is a non-negativefunction defined on .0;1/, then

�Z 1

0

�Z t

0

f .u/ du

�q

t�r dt

t

�1=q

� q

r

�Z 1

0

tf .t/�q

t�r dt

t

�1=q

(1.7.2)

and

�Z 1

0

�Z 1

tf .u/ du

�q

tr dt

t

�1=q

� q

r

�Z 1

0

tf .t/�q

tr dt

t

�1=q

: (1.7.3)

Proof Observe that the measure

d WD r

qt�r=qur=q�1 du


is a probability measure on Œ0; t� for a fixed t. Applying Jensen’s inequality with�.x/ D jxjq, we obtain

�Z t

0

f .u/ du

�q

Dq

r

�qtr

�Z t

0

f .u/u1�r=q d

�q

�q

r

�qtrZ t

0

f .u/quq�r d

Dq

r

�q�1tr�r=q

Z t

0

f .u/quq�rCr=q�1 du:

Henceforth,

Z 1

0

�Z t

0

f .u/ du

�q

t�r�1 dt

�q

r

�q�1 Z 1

0

t�1�r=q

�Z t

0

f .u/quq�rCr=q�1 du

�

dt

Dq

r

�q�1 Z 1

0

uf .u/�q

u�rCr=q�1�Z 1

ut�1�r=q dt

�

du

Dq

r

�qZ 1

0

uf .u/�q

u�r�1 du;

which proves (1.7.2).We show that (1.7.2) implies (1.7.3). Applying (1.7.2) to the function g.u/ WD

u�2f .u�1/,Z 1

0

�Z t

0

g.u/ du

�q

t�r�1 dt Dq

r

�qZ 1

0

tg.t/�q

t�r�1 dt:

The left-hand side is equal to

Z 1

0

�Z 1

1=tf .v/ dv

�q

t�r�1 dt DZ 1

0

�Z 1

sf .v/ dv

�q

sr�1 ds;

while the right-hand side to

q

r

�qZ 1

0

tg.t/�q

t�r�1 dt Dq

r

�qZ 1

0

uf .u/�q

ur�1 du:

This finishes the proof. �It is known that the interpolation spaces of the Lp.R/ spaces are Lorentz spaces

and that the interpolation spaces of Lorentz spaces are Lorentz spaces, too.

Theorem 1.7.12 If 0 < r < 1, 0 < � < 1 and r � q � 1, then

.Lr;L1/�;q D Lp;q;1

pD 1 � �

r:

Proof First we prove the equivalence

K.t; f ;Lr;L1/ �

Z tr

0

Qf .s/r ds

!1=r

: (1.7.4)

For a fixed t take

f0.x/ WD�

f .x/ � Qf .tr/f .x/=j f .x/j; if j f .x/j > Qf .tr/I0; else.

and f1 WD f � f0. Set

E WD ˚j f j > Qf .tr/�

:

It is easy to see that jEj � tr and Qf is constant on ŒjEj; tr�. Henceforth,

K.t; f ;Lr;L1/ � k f0kr C tk f1k1

D�Z

E

�j f j � Qf .tr/�r

d�

�1=r

C t Qf .tr/

D

Z jEj

0

� Qf .s/� Qf .tr/�r

ds

!1=r

C

Z tr

0

� Qf .tr/�r

ds

!1=r

D

Z tr

0

� Qf .s/ � Qf .tr/�r

ds

!1=r

C

Z tr

0

� Qf .tr/�r

ds

!1=r

� C

Z tr

0

Qf .s/r ds

!1=r

;

which shows the first part of (1.7.4).For the converse inequality assume that f D f0 C f1 with f0 2 Lr.R/ and f1 2

L1.R/. Using the inequality

�.j f j > �0 C �1/ � �.j f0j > �0/C �.j f1j > �1/;

we obtain

Qf .s/ � Qf 0..1 � �/s/C Qf 1.�s/ .0 < � < 1/:

As Qf 1 is non-increasing, we can conclude that

Z tr

0

Qf .s/r ds

!1=r

�

Z tr

0

� Qf 0..1 � �/s/�r

ds

!1=r

C

Z tr

0

� Qf 1.�s/�r

ds

!1=r

��Z 1

0

� Qf 0..1 � �/s/�rds

�1=r

C t Qf 1.0/

D .1 � �/�1=rk f0kr C tk f1k1:

Tending with � to zero, we have finished the proof of (1.7.4).First suppose that 0 < q < 1. By (1.7.4), we have

k f k.Lr ;L1/�;qD�Z 1

0

t��K.t; f ;Lr;L1/�q dt

t

�1=q

� C

0

@

Z 1

0

t��q

Z tr

0

Qf .s/r ds

!q=rdt

t

1

A

1=q

D C

Z 1

0

t��q=r

�Z t

0

Qf .s/r ds

�q=r dt

t

!1=q

: (1.7.5)

Since r � q, we can apply Hardy’s inequality (1.7.2):

k f k.Lr ;L1/�;q� C

�Z 1

0

tq=r��q=r Qf .t/q dt

t

�1=q

D C

�Z 1

0

tq=p Qf .t/q dt

t

�1=q

D C k f kp;q :

Conversely, using (1.7.4), (1.7.5) and the fact that Qf is non-increasing, we obtain

k f k.Lr ;L1/�;q� C

Z 1

0

t��q=r

�Z t

0

Qf .s/r ds

�q=r dt

t

!1=q

� C

�Z 1

0

tq=r��q=r Qf .t/q dt

t

�1=q

D C k f kp;q :

For q D 1, we have

k f k.Lr ;L1/�;1D sup

t>0t��K.t; f ;Lr;L1/

� C supt>0

t��

Z tr

0

Qf .s/r ds

!1=r

D C supt>0

t��=r

�Z t

0

Qf .s/rsr=ps�r=p ds

�1=r

(1.7.6)

� C k f kLp;1supt>0

t��=r

�Z t

0

s�r=p ds

�1=r

D C k f kLp;1;

because r � p. On the other hand, by (1.7.6),

k f k.Lr ;L1/�;q� C sup

t>0t��=r

�Z t

0

Qf .s/r ds

�1=r

� C supt>0

t.1��/=r Qf .t/

D C k f kp;q :

The proof of the theorem is complete. �Applying the reiteration theorem, we get the following general result.

Corollary 1.7.13 Suppose that 0 < � < 1 and 0 < p0; p1; q0; q1; q � 1. If p0 ¤p1, then

�

Lp0;q0 ;Lp1;q1

�

�;q D Lp;q;1

pD 1 � �

p0C �

p1:

In a special case,

�

Lp0 ;Lp1

�

�;pD Lp;

1

pD 1 � �

p0C �

p1:

Proof Let 0 < r � p0; p1; q0; q1; q and

1=pi D .1 � �i/=r .i D 0; 1/; � D .1 � �/�0 C ��1:

Notice that 1=p D .1 � �/=r. If p0 ¤ p1, then Theorems 1.7.9 and 1.7.12 imply

�

Lp0;q0 ;Lp1;q1

�

�;qD

.Lr;L1/�0;q0 ; .Lr;L1/�1;q1�

�;q

D .Lr;L1/�;q D Lp;q;

which shows the corollary. �The next lemma is an extension of Hardy’s inequality to all 0 < q < 1 and was

proved by Riviere and Sagher [274].

Lemma 1.7.14 If f � 0 is a non-increasing function on .0;1/ and 0 < q � 1,0 < s < q, then

�Z 1

0

�

1

t

Z t

0

f .u/ du

�q

ts dt

t

�1=q

� Cq;s

�Z 1

0

f .t/qts dt

t

�1=q

: (1.7.7)

Proof Consider the sublinear operator

T f .t/ WD 1

t

Z t

0

f .u/ du:

Hölder’s inequality implies

T f .t/ � t�1=pk f kp:

Hence

.T f /Q.t/ � t�1=pk f kp

and so

T W Lp �! Lp;1 .1 � p � 1/

is bounded. The operator T is clearly linear. Applying Theorem 1.7.8 and Corol-lary 1.7.13, we get that

T W .L1;L1/�;q �! .L1;1;L1/�;q

and thus

T W Lp;q �! Lp;q .1 < p < 1; 0 < q � 1/ (1.7.8)

is also bounded. As f is non-increasing, so is 1=vR v

0f .u/ du. Thus Qf .t/ D f .t/

and the non-increasing rearrangement of the function 1=vR v

0f .u/ du at a point t

is 1=tR t0

f .u/ du. Therefore, (1.7.8) yields that

�Z 1

0

�

1

t

Z t

0

f .u/ du

�q

tq=p dt

t

�1=q

� Cq;p

�Z 1

0

f .t/qtq=p dt

t

�1=q

;

which is exactly the desired inequality. �The proof of the following theorem is based on the atomic decomposition.

Theorem 1.7.15 Let f 2 Hp.R/, y > 0 and fix 0 ygf p� d�

!1=p

:

Proof Choose N 2 Z such that 2N�1 < y � 2N . Set

g WDNX

kD�1

X

i

ki ak

i

and

h WD1X

kDNC1

X

i

ki ak

i ;

where the real numbers ki and p-atoms ak

i are defined in Theorem 1.6.10. Thenf D g C h and inequality (1.6.21) implies that

jgj � C02NC1 � Cy;

which is the first inequality of the theorem.On the other hand, by (1.6.16) and (1.6.24),

khkpHp

� Cp

1X

kDNC1

X

i

ˇ

ˇ ki

ˇ

ˇ

p

D Cp

1X

kDNC12kpˇ

ˇf f� > 2kgˇˇ

D Cp

1X

kDNC1.2p/k

ˇ

ˇf.2p/k < f p� � .2p/kC1gˇˇ

� Cp

Z

f f�>2NC1g

f p� d�

� Cp

Z

f f�>ygf p� d�;

which proves the theorem. �Now the interpolation spaces between the Hardy spaces Hp.R/ can be identified.

Theorem 1.7.16 If 0 < � < 1, 0 < p0 � 1 and 0 < q � 1, then

�

Hp0 ;H1�

�;qD Hp;q ;

1

pD 1 � �

p0:

Proof Note that H1.R/ D L1.R/. Let f 2 Hp;q.R/ and Qf � be the non-increasingrearrangement of f�. Choose y in Theorem 1.7.15 such that, for a fixed t 2Œ0; 1�, y D Qf �.tp0 /. For this y let us denote the two tempered distributions inTheorem 1.7.15 by gt and ht. By the definition of the functional K,

K.t; f ;Hp0 ;H1/ � khtkHp0C t kgtkH1

:

By Theorem 1.7.15 we get that

khtkHp0� C

Z

f f�>Qf �.tp0 /g

f p0� d�

!1=p0

D C

Z tp0

0

Qf �.x/p0 dx

!1=p0

:

Consequently, for 0 < q < 1,

Z 1

0

t�� khtkHp0

�q dt

t� C

Z 1

0

t��q

Z tp0

0

Qf �.x/p0 dx

!q=p0dt

t

D CZ 1

0

t.1��/q=p0

�

1

t

Z t

0

Qf �.x/p0 dx

�q=p0 dt

t:

Using inequality (1.7.7) we obtain

Z 1

0

t�� khtkHp0

�q dt

t� C

Z 1

0

t.1��/q=p0 Qf �.t/q dt

tD C k f�kq

p;q :


Furthermore,

Z 1

0

�

t1�� kgtkH1

�q dt

t� C

Z 1

0

t.1��/q Qf �.tp0 /

q dt

t:

Substituting u D tp0 , we can see that

Z 1

0

�

t1�� kgtkH1

�q dt

t� C

Z 1

0

u.1��/q=p0 Qf �.u/q du

uD C k f�kq

p;q :

Henceforth,

k f k.Hp0 ;H1/�;qD�Z 1

0

t��K.t; f ;Hp0 ;H1/�q dt

t

�1=q

� C k f kHp;q:

If q D 1, then

supt>0

t�� khtkHp0� C sup

t>0t��

Z tp0

0

Qf �.x/p0 dx

!1=p0

D C supt>0

t��=p0

�Z t

0

Qf �.x/p0xp0=px�p0=p dx

�1=p0

� C k f�kp;1 supt>0

t��=p0

�Z t

0

x�p0=p dx

�1=p0

D C k f�kp;1

and

supt>0

t1�� kgtkH1

� C supt>0

t1�� Qf �.tp0 / � C sup

t>0t.1��/=p0 Qf �.t/ � C k f�kp;1 :

Then

k f k.Hp0 ;H1/�;1D sup

t>0t��K.t; f ;Hp0 ;H1/ � C k f kHp;1

:

To prove the converse, consider the sublinear operator T W f 7! f�. By thedefinition of the Hardy spaces,

T W H1 �! L1 and T W Hp0 �! Lp0

are bounded. Therefore, by Theorems 1.7.8 and 1.7.12,

T W �Hp0 ;H1�

�;q�! �

Lp0 ;L1�

�;qD Lp;q

is bounded, too, for all 0 < q � 1. That is to say, f 2 �Hp0 ;H1�

�;q implies

k f kHp;qD k f�kp;q � C k f k.Hp0 ;H1/�;q

:

This completes the proof of the theorem. �Applying the reiteration theorem we get the following result.

Corollary 1.7.17 Suppose that 0 < � < 1 and 0 < p0; p1; q0; q1; q � 1. If p0 ¤p1, then

�

Hp0;q0 ;Hp1;q1

�

�;qD Hp;q;

1

pD 1 � �

p0C �

p1:

In a special case,

�

Hp0 ;Hp1

�

�;pD Hp;

1

pD 1 � �

p0C �

p1

and, for 1 < p1 � 1,

�

H1;Lp1

�

�;pD Lp;

1

pD 1 � �C �

p1:

The interpolation spaces between the Hardy spaces were identified first byFefferman, Riviere and Sagher [109, 274].

As a consequence of the above results we get the following interpolationtheorem concerning Hardy-Lorentz spaces, which will be used several times in thismonograph.

Corollary 1.7.18 If a sublinear or linear operator V is bounded from Hp0 .R/ toLp0 .R/ (resp. to Hp0 .R/) and from Lp1 .R/ to Lp1 .R/ .p0 � 1 < p1 � 1/, then it isalso bounded from Hp;q.R/ to Lp;q.R/ (resp. to Hp;q.R/) for each p0 < p < p1 and0 < q � 1.

Proof If the operator V is linear, then the result follows from Theorem 1.7.8 andCorollary 1.7.17. If V is sublinear, then for a D a0 C a1, a0 2 Hp0 .R/, a1 2 Lp1 .R/,we have

jVaj � jVa0j C jVa1j:

Choose bi 2 Lpi.R/ .i D 0; 1/ such that jVaj D b0 C b1 and 0 � bi � jVaij. Then

kbikpi� kVaikpi

� Cpi kaikHpi;

which shows that the operator jVj is quasilinear. The corollary follows fromTheorem 1.7.8. �

1.8 Bounded Operators on Hardy Spaces

It is an important problem in harmonic analysis and in summability theory as towhether a linear or sublinear operator V is bounded from the Hardy space Hp.R/

to Lp.R/. If we know this boundedness for at least one p with p < 1 and for atleast one p with p > 1, then we obtain by interpolation that V is of weak type.1; 1/, which is a basic inequality in summability theory. The following sufficientcondition is used several times in the literature. If the Lp-norms of Va are uniformlybounded, where a denotes an arbitrary p-atom, then V is bounded from Hp.R/ toLp.R/ .0 < p � 1/. A typical proof of this theorem is as follows. Usually, we takean atomic decomposition

f D1X

kD0 kak;

where each ak is a p-atom and

1X

kD0j kjp

!1=p

� Cp k f kHp:

Then

jVf j �1X

kD0j kj jVakj (1.8.1)

and

kVf kpp �

1X

kD0j kjp kVakkp

p � Cp k f kpHp:

The problem is that this proof is false because the inequality (1.8.1) does notnecessarily hold. Indeed, starting from a counterexample of Meyer et al. [247],Bownik [38] have given an operator V for which (1.8.1) does not hold (see alsoBownik et al. [40, 41]). Moreover, though the Lp-norms of Va are uniformlybounded, V is not bounded from Hp.R/ to Lp.R/. For similar results see Meda et al.[245, 246], Ricci and Verdera [270] and Yang and Zhou [393]. Here we will correctthe preceding proof.

Theorem 1.8.1 Suppose that 0 < p � 1 < q < 1 and V W Lq.R/ ! Lq.R/ is abounded linear operator such that

Z

RnrIjVajp d� � Cp (1.8.2)

1.8 Bounded Operators on Hardy Spaces 61

for all .p; q/-atoms a and for some fixed r 2 N, where the interval I is the supportof the atom. Then V can uniquely be extended to a bounded linear operator fromHp.R/ to Lp.R/.

Proof Suppose that a is a .p; q/-atom with support I. By the boundedness of V andby (1.8.2), we obtain

Z

R

jVajp d� DZ

rIjVajp d�C

Z

RnrIjVajp d� (1.8.3)

��Z

rIjVajq d�

�p=q

jrIj1�p=q C Cp

� Cp

�Z

rIjajq d�

�p=q

jIj1�p=q C Cp

� Cp

jIj1�q=p�p=qjIj1�p=q C Cp (1.8.4)

D Cp:

Assume that f 2 Lq.R/\ Hp.R/ and let us take the usual atomic decomposition

f D1X

kD0 kak a.e.

as in (1.6.23), where each ak is a p-atom. As we mentioned in Remark 1.6.11, theconvergence holds also in the Lq.R/-norm. Since V is bounded on Lq.R/,

Vf D1X

kD0 kVak in the Lq-norm:

Thus

jVf j �1X

kD0j kjjVakj

and

kVf kpp �

1X

kD0j kjp kVakkp

p � Cp k f kpHp

. f 2 Lq.R/\ Hp.R//:

Since Lq.R/ \ Hp.R/ is dense in Hp.R/, V can uniquely be extended to a boundedlinear operator from Hp.R/ to Lp.R/. �

If q D 1, then the preceding proof does not work. The next theorem is usedmany times in summability theory. With the help of this result almost everywhereconvergence of various summability methods can be proved. In the proof we willextend the operator and show that the extended operator is the same as the originalone for integrable functions. The next lemma is needed for the proof.

Lemma 1.8.2 Suppose that ft .t 2 RC/ is a measurable function such that

supt2F

j ftj

p

� b < 1

for each finite subset F � RC, where 0 < p < 1. Then there exists a measurablefunction Qf t .t 2 RC/ that is almost everywhere equal to ft such that

supt2RC

ˇ

ˇ Qf t

ˇ

ˇ

p

� b:

Proof Let

a WD supF

supt2F

j ftj

p

� b;

where the supremum is taken for all finite subsets F � RC. There exists a sequenceFn for which

limn!1

supt2Fn

j ftj

p

D a:

We may suppose that Fn � FnC1 for all n 2 N. Set

g WD supn2N

supt2Fn

j ftj :

By the theorem of Beppo-Levi, kgkp D a. Clearly,

sup

j fsj ; sup0�k�n

supt2Fk

j ftj!

p

� a;

which implies that

ksup .j fsj ; g/kp � a D kgkp;

in other words j fsj � g almost everywhere for all s > 0. �

For operators Vt W L1.R/ ! L1.R/ .t > 0/ let

V� f WD supt>0

jVt f j:

More precisely, under V� f we understand a function

suptn>0;n2N

jVtn f j

for some .tn; n 2 N/ which satisfies that

jVt f j � suptn>0;n2N

jVtn f j a.e.

for all t > 0. Under the conditions of Lemma 1.8.2, such a function does exist. ThusV� f is measurable.

Theorem 1.8.3 For each t > 0 let Vt W L1.R/ ! L1.R/ be a bounded linearoperator. Suppose that

Z

RnrIjV�ajp0 d� � Cp0

for all .p0; q/-atoms a and for some fixed r 2 N and 0 < p0 � 1, where theinterval I is the support of the atom. If V� is bounded from Lp1 .R/ to Lp1 .R/ forsome 1 < p1 � q � 1, then

kV� f kp � Cp k f kHp. f 2 Hp.R/\ L1.R// (1.8.5)

for all p0 � p � p1. If

limk!1 fk D f in the Hp-norm implies that lim

k!1 Vt fk D Vt f in S0.R/

for all t > 0, then (1.8.5) holds for all f 2 Hp.R/.

Proof By Lemma 1.8.2, it is enough to prove the theorem for

VF f WD supt2F

jVt f j

for every finite subset F � RC. Similar to the inequality (1.8.3),

Z

R

jVFajp0 d� ��Z

rIjVFajp1 d�

�p0=p1

jrIj1�p0=p1 C Cp0

� Cp0

�Z

rIjajp1 d�

�p0=p1

jIj1�p0=p1 C Cp0

� Cp0

�Z

rIjajq d�

�p0=q

jIj1�p0=q C Cp0

� Cp0

jIj1�q=p0�p0=qjIj1�p0=q C Cp0

D Cp0 :

Let us take again the atomic decomposition (1.6.23):

f D1X

kD0 kak in S0.R/ and

1X

kD0j kjp0

!1=p0

� Cp0k f kHp0;

where the convergence holds also in the L1.R/-norm, if f 2 H1.R/\ Hp0 .R/. SinceVt is bounded on L1.R/, we have

Vt f D1X

kD0 kVtak

and

jVF f j �1X

kD0j kj jVFakj

for f 2 H1.R/. Thus

kVF f kp0p0 �

1X

kD0j kjp0 kVFakkp0

p0 � Cp0 k f kp0Hp0

(1.8.6)

for all f 2 H1.R/ \ Hp0 .R/. This and interpolation (Corollary 1.7.18) prove thetheorem if p0 D 1. Assume that p0 < 1. Since H1.R/ is dense in L1.R/ as well asin Hp0 .R/, we can extend uniquely the operator VF such that (1.8.6) holds for allf 2 Hp0 .R/. Let us denote this extended operator by V 0

F. It is easy to see that V 0F is

sublinear. Then VF f D V 0F f for all f 2 H1.R/ \ Hp0 .R/. It is enough to show that

this equality holds for all f 2 Hp0 .R/\ L1.R/. From (1.8.6), we get by interpolationthat the operator

V 0F is bounded from Hp;1.R/ to Lp;1.R/ (1.8.7)

when p0 0

� �.ˇ

ˇV 0F fˇ

ˇ > �/ D

V 0F f

1;1 � C k f kH1;1 � C k f k1 (1.8.8)

for all f 2 L1.R/. Let f 2 L1.R/ and fk 2 H1.R/ \ Hp0 .R/ such that limk!1 fk D fin the L1.R/-norm. Since Vt is bounded on L1.R/, the inequality

sup�>0

� ��ˇ

ˇV 0F f � VF f

ˇ

ˇ > �� sup

�>0

� ��ˇ

ˇV 0F f � V 0

F fkˇ

ˇ > �=2�

C sup�>0

� � .jVF fk � VF f j > �=2/

� sup�>0

� ��ˇ

ˇV 0F. f � fk/

ˇ

ˇ > �=2�

CX

t2F

sup�>0

� � .jVt. fk � f /j > �=.2jFj//

� C k f � fkk1 ! 0

as k ! 1, shows that

V 0F f D VF f for all f 2 L1.R/:

Here jFj denotes the cardinality of the finite set F. Consequently, (1.8.8) holds alsofor VF and (1.8.6) for all f 2 L1.R/.

Now assume that Vt is defined also for tempered distributions and thatlimk!1 fk D f in the Hp.R/-norm implies limk!1 Vt fk D Vt f in the sense oftempered distributions .t > 0/. Suppose that p < 1 and fk 2 Hp.R/\L1.R/ .k 2 N/.Since by (1.8.5), Vt fk is convergent in the Lp.R/-norm as k ! 1, we can identifythe distribution Vt f with the Lp.R/-limit limk!1 Vt fk. Hence the same holds forVF f :

VF f D limk!1 VF fk in the Lp.R/-norm:

Consequently, (1.8.5) holds for all f 2 Hp.R/. The proof of the theorem iscomplete. �

The next corollary was already shown in the preceding proof.

Corollary 1.8.4 For each t > 0 let Vt W L1.R/ ! L1.R/ be a bounded linearoperator. Suppose that

Z

RnrIjV�ajp0 d� � Cp0

for all .p0; q/-atoms a and for some fixed r 2 N and 0 < p0 < 1, where theinterval I is the support of the atom. If V� is bounded from Lp1 .R/ to Lp1 .R/ forsome 1 < p1 � q � 1, then

kV� f k1;1 D sup�>0

� �.jV� f j > �/ � C k f k1 . f 2 L1.R//: (1.8.9)

In summability theory we will apply this theorem for q D 1 and for operators

Vtf .x/ WD f � Kt.x/ WDZ

R

f .t/Kt.x � t/ dt .t > 0/;

where Kt 2 L1.R/ are summability kernels and f 2 L1.R/. We omit the indices tand write for the operators and summability kernels simply V and K. Obviously, theconvolution is well defined for all K 2 L1.R/ and f 2 Lp.R/ .1 � p � 1/, but wehave extended it to bounded tempered distributions in Definition 1.4.8.

Theorem 1.8.5 Suppose that Vf WD f � K for all bounded tempered distributions,where K 2 L1.R/. If 0 < p < 1 and

limk!1 fk D f in the Hp-norm, then lim

k!1 Vfk D Vf in S0.R/:

Proof First we show that Vf D f � K is a tempered distribution for each f 2 Hp.R/.To this end we have to show that if limk!1 hk D h in S.R/, then

limk!1 f � K.hk/ D f � K.h/;

or, by Definition 1.4.8,

limk!1

Z

R

. f � Mhk/.x/ MK.x/ dx DZ

R

. f � Mh/.x/ MK.x/ dx: (1.8.10)

Suppose that h 2 S.R/ with khkKm � 1=2. Then

khkkKm

D supx2R;˛�m

.1C jxj/mC1jh.˛/k .x/j

� supx2R;˛�m

.1C jxj/mC1j.h.˛/k � h.˛//.x/j C supx2R;˛�m

.1C jxj/mC1jh.˛/.x/j

� 1;

when k is large enough, because the first summand tends to 0. Then for such a k,

ˇ

ˇ

ˇ. f � Mhk/.x/ˇ

ˇ

ˇ � f�.y/ for every y with jx � yj � 1;

thus

ˇ

ˇ

ˇ. f � Mhk/.x/ˇ

ˇ

ˇ

p � 1

jI.x; 1/jZ

I.x;1/f�.y/

p dy � 1

2k f kp

Hp:

This shows that . f � Mhk/.x/ is uniformly bounded in k, when f 2 Hp.R/. Moreover,since Txhk ! Txh in S.R/, we have

. f � Mhk/.x/ D f .Tx Mhk/ ! f .Tx Mh/ D . f � Mh/.x/ .x 2 R/:

Now Lebesgue’s theorem implies (1.8.10) and Vf is indeed a tempered distribution.Suppose that limk!1 fk D f in the Hp.R/-norm. We have to show that

limk!1 fk � K.h/ D f � K.h/

for all h 2 S.R/, or, in other words,

limk!1

Z

R

. fk � Mh/.x/ MK.x/ dx DZ

R

. f � Mh/.x/ MK.x/ dx: (1.8.11)

Thenˇ

ˇ

ˇ. fk � Mh/.x/ˇ

ˇ

ˇ � C k fkkHp� Ck f kHp

and

. fk � Mh/.x/ D fk.Tx Mh/ ! f .Tx Mh/ D . f � Mh/.x/ .x 2 R/

imply (1.8.11). �We can formulate a weak version of Theorem 1.8.3 as follows.

Lemma 1.8.6 Suppose that ft .t 2 RC/ is a measurable function such that

supt2F

j ftj

p;1� b < 1

for each finite subset F � RC, where 0 ��

� bp

for a fixed � > 0, we can finish the proof as in Lemma 1.8.2. �

For a set H, we use the notation

�.j f j > �;H/ WD �.1Hj f j > �/:

Theorem 1.8.7 For each t > 0 let Vt W L1.R/ ! L1.R/ be a bounded linearoperator. Suppose that

sup�>0

�p�.jV�aj > �;R n rI/ � Cp

for all .p; q/-atoms a and for some fixed r 2 N and 0 < p < 1, where the intervalI is the support of the atom. If V� is bounded from Lp1 .R/ to Lp1 .R/ for some 1 <p1 � q � 1, then

kV� f kp;1 � Cp k f kHp. f 2 Hp.R/\ L1.R//: (1.8.12)

If


k!1 Vt fk D Vt f in S0.R/

for all t > 0, then (1.8.12) holds for all f 2 Hp.R/.

Proof By Lemma 1.8.6, it is enough to prove the theorem again for VF f , with everyfinite subset F � RC. It is easy to see that

sup�>0

�p�.jVFaj > �; rI/ �Z

rIjVFajp d�

� Cp

�Z

R

jVFajp1 d�

�p=p1

jIj1�p=p1

� Cp:

Thus

sup�>0

�p�.jVFaj > �/ � Cp (1.8.13)

for every .p; q/-atom a.Suppose f 2 Hp.R/ has an atomic decomposition of the form (1.6.15),

f D1X

kD0 kak:


For � > 0 set

gk WD jVFakj1fjVFakj��=j k jg; hk WD jVFakj1fjVFakj>�=j k jg

and

E� WD1[

kD0fhk ¤ 0g:

Since by (1.8.13),

jfhk ¤ 0gj Dˇ

ˇ

ˇ

ˇ

�

jVFakj > �

j kj� ˇ

ˇ

ˇ

ˇ

� Cpj kjp

�p;

we have

jE�j � Cp��p

1X

kD0j kjp:

Moreover,

kgkk1 DZ

fjVFakj��=j k jgjVFakj d�

�Z �=j k j

0

�.jVFakj > t/ dt

� Cp

�

�

j kj�1�p

;

which implies

ˇ

ˇ

ˇ

ˇ

ˇ

( 1X

kD0j kjjVFakj > �

) ˇ

ˇ

ˇ

ˇ

ˇ

� jE�j Cˇ

ˇ

ˇ

ˇ

ˇ

(

x 62 E� W1X

kD0j kjjVFak.x/j > �

) ˇ

ˇ

ˇ

ˇ

ˇ

� jE�j Cˇ

ˇ

ˇ

ˇ

ˇ

( 1X

kD0j kjgk > �

) ˇ

ˇ

ˇ

ˇ

ˇ

� jE�j C ��11X

kD0j kj kgkk1

� Cp��p

1X

kD0j kjp:


Using the inequality

jVF f j �1X

kD0j kjjVFakj;

we have proved the theorem for all f 2 H1.R/ \ Hp.R/. The extension to f 2L1.R/ \ Hp.R/ can be done as in Theorem 1.8.3. �

Note that the weak type inequality (1.8.9) follows also from this theorem byinterpolation.

Chapter 2One-Dimensional Fourier Transforms

In this chapter, we study the theory of one-dimensional Fourier transforms, theinversion formula, convergence and summability of Fourier transforms. In the firsttwo sections, we introduce the Fourier transform for Schwartz functions and weextend it to L2.R/, L1.R/, Lp.R/ .1 � p � 2/ functions as well as to tempereddistributions. We prove some elementary properties and the inversion formula. InSect. 2.4, we deal with the convergence of Dirichlet integrals. Using some resultsfor the partial sums of Fourier series proved in Sect. 2.3, we show that the Dirichletintegrals converge in the Lp.R/-norm to the function .1 < p < 1/. The proof ofCarleson’s theorem, i.e. that of the almost everywhere convergence can be foundin Carleson [52], Grafakos [152], Arias de Reyna [8], Muscalu and Schlag [253],Lacey [207] or Demeter [88].

It was proved by Fejér [116] that the Fejér means of the one-dimensional Fourierseries of a continuous function converge uniformly to the function. The sameproblem for integrable functions was investigated by Lebesgue [212]. He provedthat for every integrable function f ,

1

n C 1

nX

kD0sk f .x/ ! f .x/ as n ! 1

at each Lebesgue point of f , where sk f denotes the kth partial sum of the Fourierseries of f . Almost every point is a Lebesgue point of f .

Later, Riesz [271], Butzer and Nessel [46], Stein and Weiss [311] and Torchinsky[330] proved the same convergence result for the Riesz, Weierstrass, Picard,Bessel and de La Vallée-Poussin summations. In Sect. 2.5, we consider a generalsummability method, the so-called �-summability, which is generated by a singlefunction � and which includes all the above well-known summability methods. Inthe next section, we introduce the Wiener amalgam spaces and prove the normconvergence of the �-summation if the Fourier transform of � is integrable. In


71

72 2 One-Dimensional Fourier Transforms

Sects. 2.7 and 2.9, we study the almost everywhere convergence of the �-means offunctions from the Wiener amalgam spaces. It is shown that the �-means convergeto f at each Lebesgue point of f if and only if the Fourier transform of � is in a Herzspace. In Sect. 2.8, we prove that the maximal operator of the �-means is boundedfrom the Hardy space Hp.R/ to Lp.R/ for p0 < p � 1, where p0 is depending on thesummability function � . We deal with strong summability of Fourier transforms andpresent, amongst others, the analogue of Gabisonya’s result in Sect. 2.10. Finally, asspecial cases of the �-summability, some special examples are presented.

2.1 Fourier Transforms

Definition 2.1.1 Let f 2 S.R/. The Fourier transform of f is defined by

F f .�/ WDbf .�/ WD 1p2�

Z 1

�1f .t/ e�{t� dt .� 2 R/ :

We use the notation { D p�1. Let us see an important example for the Fouriertransform.

Proposition 2.1.2 The Fourier transform of f .x/ D e�x2=2 is the same function, i.e.bf .�/ D e��2=2.

Proof By the definition of the Fourier transform,

bf .�/ D 1p2�

Z 1

�1e�x2=2e�{x� dx D e��2=2

p2�

Z 1

�1e�.xC{�/2=2 dx:

Observe that the function on the right-hand side,

s 7!Z 1

�1e�.xC{s/2=2 dx

is constant. Indeed, its derivative is equal to

Z 1

�1e�.xC{s/2=2.�x � {s/{ dx D {

Z 1

�1d

dxe�.xC{s/2=2 dx D 0:

Writing s D 0, we have

bf .�/ D e��2=2p2�

Z 1

�1e�x2=2 dx:

2.1 Fourier Transforms 73

Observe that

�Z 1

�1e�x2=2 dx

�2

DZ 1

�1

Z 1

�1e�x2=2�y2=2 dx dy

DZ 2�

0

Z 1

0

re�r2=2 d� dr

D 2�

Z 1

0

re�r2=2 dr

D 2�;

thusbf .�/ D e��2=2. �

Definition 2.1.3 Let f be an arbitrary function, x; s; t; ! 2 R, s ¤ 0. Then thetranslation, the dilation and the modulation of f is defined by

Tx f .t/ WD f .t � x/; Dsf .t/ WD jsj�1=2 f .s�1t/; M! f .t/ WD ei!tf .t/;

respectively. The involution function is given by

f �.x/ WD f .�x/:

The Fourier transform of these operators is characterized in the next theorem.

Theorem 2.1.4 Let f ; g 2 S.R/, x; ! 2 R, s > 0, ˇ 2 N and b 2 C. Then

(a) bf 2 L1.R/ and

bf

1 � k f k1=p2� ,

(b) 1f C g Dbf Cbg,(c) bbf D bbf ,(d) bTx f D M�xbf ,(e) bM! f D T!bf ,(f) bDsf D Ds�1bf ,

(g) bf � Dbf ,

(h)

bf .ˇ/�

.�/ D .{�/ˇbf .�/,

(i)

bf�.ˇ/

.�/ D .�{/ˇ

1tˇf .t/�

.�/,

(j) bf 2 S.R/,(k) bf � g D p

2�bf �bg.

Proof (a) follows from

bf

1 D 1p2�

sup�2R

ˇ

ˇ

ˇ

ˇ

Z

R

f .t/e�{t� dt

ˇ

ˇ

ˇ

ˇ

� 1p2�

Z

R

j f .t/j dt:

(b) and (c) are clear from the definition. (d)–(g) hold because of

bTx f .�/ D 1p2�

Z

R

f .t � x/ e�{t� dt D 1p2�

Z

R

f .u/ e�{.uCx/� du D M�xbf .�/;

bM! f .�/ D 1p2�

Z

R

M! f .t/ e�{t� dt D 1p2�

Z

R

f .t/ e�{t.��!/ dt D T!bf .�/;

bDs f .�/ D jsj�1=2p2�

Z

R

f .s�1t/ e�{t� dt D jsj1=2p2�

Z

R

f .y/ e�{ys� dy D Ds�1bf .�/

and

bf �.�/ D 1p2�

Z

R

f .�t/ e�{t� dt D 1p2�

Z

R

f .�t/ e{t� dt Dbf .�/:

For (h) let first ˇ D 1. By integrating by parts,

p2�b. f 0/.�/ D

Z

R

f 0.t/ e�{t� dt

D �

f .t/ e�{t� 1�1 �

Z

R

f .t/ .�{�/ e�{t� dt

D p2� .{�/bf .�/:

Applying this integration ˇ-times, we get (h). (i) follows by repeating the differen-tiation ˇ-times:

bf�0.�/ D 1p

2�

Z

R

f .t/ e�{t� .�{t/ dt D �{

btf .t/�

.�/:

Applying (i), we conclude

sup�2R

ˇ

ˇ

ˇ

ˇ

�˛

bf�.ˇ/

.�/

ˇ

ˇ

ˇ

ˇ

D sup�2R

ˇ

ˇ

ˇ.{�/˛

1tˇf .t/�

.�/ˇ

ˇ

ˇ

and so by (h),

sup�2R

ˇ

ˇ

ˇ

ˇ

�˛

bf�.ˇ/

.�/

ˇ

ˇ

ˇ

ˇ

D sup�2R

ˇ

ˇ

ˇ

ˇ

ˇ

�

tˇf .t/�.˛/

�b

.�/

ˇ

ˇ

ˇ

ˇ

ˇ

�Z

R

ˇ

ˇ

ˇ

�

tˇf .t/�.˛/

ˇ

ˇ

ˇ dt < 1


for all ˛; ˇ 2 N, since t 7! �

tˇf .t/�.˛/ 2 S.R/ � L1.R/. Hencebf 2 S.R/. Finally,

bf � g.�/ D 1p2�

Z

R

�Z

R

f .x � t/g.t/ dt

�

e�{x� dx

D 1p2�

Z

R

�Z

R

f .x � t/ e�{.x�t/� dx

�

g.t/ e�{t� dt

D 1p2�

Z

R

�Z

R

f .u/ e�2�{u� du

�

g.t/ e�{t� dt

D p2�bf .�/ �bg.�/;

which finishes the proof of the theorem. �Modifying slightly the definition of the Fourier transform, we obtain the inverse

Fourier transform.

Definition 2.1.5 The inverse Fourier transform of f 2 S.R/ is defined by

f _.�/ WDbf .��/ D 1p2�

Z 1

�1f .t/ e{t� dt .� 2 R/ :

Clearly the analogous properties of Theorem 2.1.4 remain true for the inverseFourier transform. The following theorem deals with the connection between theFourier transform and the inverse Fourier transform.

Theorem 2.1.6 If f ; g; h 2 S.R/, then

.a/Z 1

�1f .x/bg.x/ dx D

Z 1

�1bf .x/g.x/ dx;

.b/

bf�_ D

2

�

f _�

D f ;

.c/Z 1

�1f .x/h.x/ dx D

Z 1

�1bf .t/bh.t/ dt;

.d/ k f k2 D

bf

2D

f _

2:

Proof By Fubini’s theorem,Z 1

�1f .x/bg.x/ dx D 1p

2�

Z 1

�1f .x/

Z 1

�1g.t/e�{tx dt dx

D 1p2�

Z 1

�1

Z 1

�1f .x/e�{tx dx g.t/ dt

DZ 1

�1bf .t/g.t/ dt;


which is exactly (a). For (b), we consider the function

g.x/ WD e{xt e�.�x/2=2 D ��1=2MtD 1�e�x2=2 .� > 0; t 2 R/

and its Fourier transform,

bg.x/ D ��1=2TtD�

�

1

e�x2=2

�

D ��1=2TtD�e�x2=2 D 1

�e�.x�t/2=2�2 :

We apply (a) for the function g:

Z 1

�1f .x/

1

�e�.x�t/2=2�2 dx D

Z 1

�1bf .x/ e{xt e�.�x/2=2 dx: (2.1.1)

Tending with � to 0, we can apply Lebesgue’s theorem on the right-hand side.Indeed,bf 2 L1.R/ since f 2 S.R/ andbf 2 S.R/ and so

lim�!0

Z 1

�1bf .x/ e{xt e�.�x/2=2 dx D

Z 1

�1bf .x/ e{xt dx D p

2�

bf�_.t/:

We will prove that the left-hand side of (2.1.1) tends top2�f .t/ as � ! 0.

Indeed,

Z 1

�1f .x/

1

�e�.x�t/2=2�2 dx D

Z 1

�1f .t � y/

1

�e�y2=2�2 dy:

SinceZ 1

�11

�e�y2=2�2 dy D

Z 1

�1e�x2=2 dx D p

2�; (2.1.2)

we concludeˇ

ˇ

ˇ

ˇ

Z 1

�1f .t � y/

1

�e�y2=2�2 dy � p

2�f .t/

ˇ

ˇ

ˇ

ˇ

�Z 1

�1

ˇ

ˇ

ˇ f .t � y/ � f .t/ˇ

ˇ

ˇ

1

�e�y2=2�2 dy

DZ

jyj�ı

ˇ

ˇ

ˇ f .t � y/� f .t/ˇ

ˇ

ˇ

1

�e�y2=2�2 dy

CZ

jyj>ı

ˇ

ˇ

ˇ f .t � y/ � f .t/ˇ

ˇ

ˇ

1

�e�y2=2�2 dy

D .I/C .II/ :

If a continuous function f converges to a 0 at ˙1, then f is uniformlycontinuous. So every Schwartz function is uniformly continuous. For each � > 0

there exists ı > 0 such that

j f .t � y/� f .t/j < � if jyj < ı and t 2 R:

By (2.1.2),

.I/ < �Z

jyj�ı1

�e�y2=2�2 dy <

p2��:

On the other hand,

.II/ � 2k f k1Z

jyj>ı1

�e�y2=2�2 dy

D 2k f k1Z

jxj>ı=�e�x2=2 dx ! 0

as � ! 0, which proves (b).

Let g Dbh. Then

g.x/ D 1p2�

Z 1

�1h.t/ e�{tx dt D 1p

2�

Z 1

�1h.t/ e{tx dt D �

h�_.x/

and sobg D h. Substituting g into (a), we obtain (c). Applying (c) to h D f , we get

the equality k f k2 D

bf

2. The other equality of (d) follows from f _.x/ Dbf .�x/.�

The equation (b) means that f _ is indeed the inverse of the Fourier transform, sowe will use the notation

F�1f WD f _:

The equation in (c) and (d) is called Plancherel theorem. Now we are extending theFourier transform to square integrable functions. We know that S.R/ � L2.R/ isdense in L2.R/. Thus there exist fn 2 S.R/ such that

limn!1 fn D f in the L2.R/-norm. (2.1.3)

Definition 2.1.7 If f 2 L2.R/, then choose functions fn 2 S.R/ with theproperty (2.1.3). Let

bf WD limn!1

bfn in the L2.R/-norm.


The definition is well defined because the sequence

bf n; n 2 N

�

is Cauchy in

L2.R/ by Plancherel theorem. Indeed,

bf n �bf m

2D kfn � fmk2 ! 0

as n;m ! 1. Thus the limitbf does exist in the L2.R/-norm. This limit is unique,since if we have two sequences . fn/ and .gn/ with property (2.1.3), thenbf n ! bfandbgn !bg. The merger of the sequences . fn/ and .gn/ satisfies also (2.1.3), so theFourier transforms of the merged sequence are also convergent in L2.R/. Thusbf Dbg. This definition is really an extension of the Fourier transform, since if f 2 S.R/,then we can choose the sequence fn D f .

The definition implies easily that

bf

2D lim

n!1

bfn

2D lim

n!1 k fnk2 D k f k2 (2.1.4)

for all f 2 L2.R/. The other parts of Theorem 2.1.6 can be shown simply.

Theorem 2.1.8 If f ; g; h 2 L2.R/, then Theorem 2.1.6 holds.

Proof Let f 2 L2.R/, g; fn 2 S.R/ and

limn!1 fn D f in the L2.R/-norm.

By Hölder’s inequality,

ˇ

ˇ

ˇ

ˇ

Z 1

�1fn.x/bg.x/� f .x/bg.x/ dx

ˇ

ˇ

ˇ

ˇ

� k fn � f k2 kbgk2 ! 0

as n ! 1, thus

limn!1

Z 1

�1fn.x/bg.x/ dx D

Z 1

�1f .x/bg.x/ dx:

Similarly,

limn!1

Z 1

�1bf n.x/g.x/ dx D

Z 1

�1bf .x/g.x/ dx;

which shows (a). The other parts can be proved in the same way. �The original Definition 2.1.1 of the Fourier transform remains true for functions

from L1.R/ \ L2.R/.


Theorem 2.1.9 If f 2 L1.R/\ L2.R/, then

bf .�/ D 1p2�

Z 1

�1f .t/ e�{t� dt a.e.

Proof First suppose that f has compact support. Then there exist functions fn 2 S.R/such that

limn!1 fn D f in the L2.R/-norm.

We may suppose that supp fn � supp f . By Hölder’s inequality, this convergenceholds also in the L1.R/-norm. By the definition,

bf D limn!1

bf n in the L2.R/-norm. (2.1.5)

On the other hand, since fn ! f in the L1.R/-norm,

limn!1

bf n.�/ D limn!1

1p2�

Z 1

�1fn.t/ e�{t� dt D 1p

2�

Z 1

�1f .t/ e�{t� dt

for almost every � 2 R. This and (2.1.5) imply

bf .�/ D 1p2�

Z 1

�1f .t/ e�{t� dt

for almost all � 2 R if f has compact support.Now let f 2 L1.R/ \ L2.R/ be arbitrary and �n WD f1Œ�n;n�. Then

limn!1�n D f in the L1.R/-norm and L2.R/-norm as well.

From this it follows that

limn!1

b�n Dbf in the L2.R/-norm,

because by (2.1.4),

limn!1

b�n �bf

2D lim

n!1 k�n � f k2 D 0:

By the first part of the proof,

b�n.�/ D 1p2�

Z 1

�1�n.t/e

�{t� dt:

Since �n ! f in the L1.R/-norm,

limn!1

b�n.�/ D 1p2�

limn!1

Z 1

�1�n.t/ e�{t� dt D 1p

2�

Z 1

�1f .t/ e�{t� dt;

which finishes the proof. �Then the Fourier transform can be extended to integrable function as follows.

Definition 2.1.10 The Fourier transform of f 2 L1.R/ is defined by

bf .�/ WD 1p2�

Z 1

�1f .t/ e�{t� dt:

It is easy to see that Theorem 2.1.4. (a) holds also for f 2 L1.R/.

Theorem 2.1.11 (Riemann-Lebesgue) If f 2 L1.R/, thenbf is uniformly continu-ous and

limj�j!1

bf .�/ D 0:

Proof First consider the characteristic function of the interval Œa; b�, f .x/ D 1Œa;b�.x/.Then

b1Œa;b�.�/ D 1p2�

Z b

ae�{�x dx D e�{�b � e�{�a

�p2�{�

and the theorem holds for 1Œa;b� and also for step functions

f .x/ DnX

iD0˛i1Ai.x/;

where ˛i 2 R and Ai � R are intervals. Since the step functions are dense in L1.R/,for all f 2 L1.R/ and � > 0 there exists a step function g such that

k f � gk1 < �:

By Theorem 2.1.4. (a),

ˇ

ˇ

ˇ

bf .�/ˇ

ˇ

ˇ �ˇ

ˇ

ˇ

bf .�/ �bg.�/ˇ

ˇ

ˇC jbg.�/j D k f � gk1 C jbg.�/j < 2�;

if j�j is large enough.

For the uniform continuity notice that

p2�ˇ

ˇ

ˇ

bf .�/ �bf .�0/ˇ

ˇ

ˇ Dˇ

ˇ

ˇ

ˇ

Z

R

f .x/e�{x� dx �Z

R

f .x/e�{x�0 dx

ˇ

ˇ

ˇ

ˇ

�Z

R

j f .x/j ˇˇe�{x.��0/ � 1ˇˇ dx:

The integrand is depending on .� � �0/, it can be estimated by 2j f .x/j, so byLebesgue’s theorem the integral tends to 0 as � ! �0. �

We can extend the Fourier transform further to Lp.R/ functions .1 < p < 2/.Recall that L1.R/ \ L2.R/ � Lp.R/ .1 < p < 2/.

Definition 2.1.12 If f 2 Lp.R/ for some 1 < p < 2, then f can be decomposed intothe sum f D f1 C f2, where f1 2 L1.R/ and f2 2 L2.R/. Let

bf D bf1 Cbf2:

The functions f1 and f2 can be chosen such that f1 D f1fj f j�1g and f2 D f1fj f j<1g.Indeed, f1 2 L1.R/ and f2 2 L2.R/ because

Z

R

j f1.x/j dx DZ

R

1fj f j�1g.x/ j f .x/j dx �Z

R

1fj f j�1g.x/ j f .x/jp dx < 1

andZ

R

j f2.x/j2 dx DZ

R

1fj f j<1g.x/ j f .x/j2 dx �Z

R

1fj f j<1g.x/ j f .x/jp dx < 1:

Note thatbf is independent of the decomposition of f , since if

f D f1 C f2 and f D h1 C h2;

f1; h1 2 L1.R/, f2; h2 2 L2.R/, then

f1 � h1 D h2 � f2 2 L1.R/\ L2.R/:

Hencebf 1 �bh1 Dbh2 �bf 2.In the next section we will extend the Fourier transform to tempered distributions.

Note that the Fourier transform of a function from Lp.R/ .2 < p < 1/ is notnecessary a function.

Now we generalize Theorems 2.1.4 (a) and 2.1.6 (d) for Lp.R/ functions.

Theorem 2.1.13 (Hausdorff-Young) If 1 � p � 2 and 1=p C 1=p0 D 1, then

F W Lp.R/ ! Lp0.R/ is bounded and

bf

p0

� C k f kp :

Proof The two theorems just mentioned imply that

F W L1.R/ ! L1.R/ and F W L2.R/ ! L2.R/

are bounded linear operators. Corollary 1.7.13 and Theorem 1.7.8 imply that

F W .L1;L2/�;p0 ! .L1;L2/�;p0

is bounded, too, where 0 < � < 1. Let

1

pD 1� �

1C �

2D 1 � �

2and

1

qD 1 � �

1 C �

2D �

2:

Thus 1=p C 1=q D 1, q D p0 and

bf

p0

� C k f kp;p0 � C k f kp ;

which completes the proof. �

2.2 Tempered Distributions

In this section we extend the Fourier transform and inverse Fourier transform totempered distributions. By Theorem 2.1.8 (a),

Z 1

�1f .x/bg.x/ dx D

Z 1

�1bf .x/g.x/ dx (2.2.1)

if f 2 S.R/ and g 2 L2.R/. This equation remains true for g 2 L1.R/ as well.Indeed, since S.R/ � L1.R/ is dense in L1.R/, there exist gn 2 S.R/ such that

limn!1 gn D g in the L1.R/-norm.

Of course (2.2.1) holds for each gn. Then

limn!1

Z 1

�1bf .x/gn.x/ dx D

Z 1

�1bf .x/g.x/ dx

becausebf is bounded. Since Theorem 2.1.4 (a) holds for g 2 L1.R/, we have

limn!1bgn Dbg in the L1.R/-norm:

2.2 Tempered Distributions 83

Then f 2 L1.R/ implies

limn!1

Z 1

�1f .x/bgn.x/ dx D

Z 1

�1f .x/bg.x/ dx:

The definition of the Fourier transform yields that (2.2.1) is valid for all g 2 Lp.R/

with 1 � p � 2.Motivated by Eq. (2.2.1), we extend the Fourier transform to tempered distribu-

tions as follows.

Definition 2.2.1 For a tempered distribution u 2 S0.R/, we define the Fouriertransform and the inverse Fourier transform by

bu. f / WD u

bf�

and u_. f / WD u�

f _� . f 2 S.R// :

Example 2.2.2 bı0 D 1 since

bı0. f / D ı0

bf�

Dbf .0/ DZ

R

f .x/ � 1 dx:

Thusbı0 D 1. If ıa. f / WD f .a/, then

bıa. f / D ıa

bf�

Dbf .a/ DZ

R

f .x/ e�2�{xa dx

and sobıa.x/ D e�2�{xa.

Definition 2.2.3 The translation, dilation and modulation of a tempered distributionis defined by

Txu. f / WD u.T�x f /; Dsu. f / WD u�

Ds�1f

�

; M!u. f / WD u .M! f / ;

where f 2 S.R/, x; s; ! 2 R, s ¤ 0.

Theorem 2.2.4 If u; v 2 S0.R/, h 2 S.R/, x; ! 2 R, s > 0 and b 2 C, then

(a) 1u C v Dbu Cbv,(b) cb u D bbu,(c) bTxu D M�xbu,(d) bM!u D T!bu,(e) bDsu D Ds�1bu,(f) bu 2 S0.R/,(g) .bu/_ D b.u_/ D u,(h) bh � u D p

2�bh �bu.


Proof Let f 2 S.R/. (a) and (b) are trivial. (c)–(e) follow from

bTxu . f / D Txu

bf�

D u

T�xbf�

D u

1M�x f�

Dbu .M�x f / D M�xbu .f / ;

bM!u . f / D M!u

bf�

D u

M!bf�

D u

1T�! f�

Dbu .T�! f / D T!bu . f /

and

bDsu . f / D Dsu

bf�

D u

Ds�1bf�

D u

bDsf�

Dbu .Dsf / D Ds�1bu . f / ;

respectively. We can see from the proof of Theorem 2.1.4 (h) that the convergencefk ! f in S.R/ impliesbf k !bf in S.R/ as k ! 1. The linearity ofbu is clear and thecontinuity comes from

bu . fk/ D u

bf k

�

! u

bf�

Dbu. f / as k ! 1:

We get (g) from

.bu/_ . f / D .bu/�

f _� D .u/

bf _�

D u. f /:

Applying Theorem 2.1.4 ( j) for inverse Fourier transforms, we have

Mh �bf�_ D

Mh�_ �

bf�_ Dbh � f :

Hence

Mh �bf D b

bh � f

and

bh � u . f / D h � u

bf�

D uMh �bf

�

D p2� u

�

b

bh � f

�

D p2�bu

bh � f�

D p2�bh �bu . f / :

The proof of the theorem is complete. �We introduce a convergence for tempered distributions.

Definition 2.2.5 Let u; uk 2 S0.R/ .k 2 N/ be tempered distributions. Then uk ! uin S0.R/ if uk. f / ! u. f / for all f 2 S.R/-re.

Recall that convergence in S.R/ implies convergence in Lp.R/ for any 1 � p �1. In the next theorem we show that convergence in Lp.R/ implies convergence inS0.R/.

2.3 Partial Sums of Fourier Series 85

Theorem 2.2.6 If uk; u 2 Lp.R/ for some 1 � p � 1 and uk ! u in the Lp.R/-norm, then uk ! u in S0.R/ as k ! 1.

Proof If f 2 S.R/, then

juk. f /� u. f /j Dˇ

ˇ

ˇ

ˇ

Z

R

f .x/ .uk.x/ � u.x// dx

ˇ

ˇ

ˇ

ˇ

� k f kp0 kuk � ukp ! 0;

as k ! 1 because f 2 Lp0.R/. �The convergence in S0.R/ is inherited for the Fourier transform of tempered

distributions.

Theorem 2.2.7 Let u; uk 2 S0.R/ .k 2 N/ be tempered distributions. If uk ! u inS0.R/, thenbuk !bu in S0.R/ as k ! 1.

Proof Sincebf 2 S.R/,

buk. f / D uk

bf�

! u

bf�

Dbu. f / . f 2 S.R// ;

which shows the result. �

2.3 Partial Sums of Fourier Series

In this section we introduce the Fourier series and prove that the partial sums areuniformly bounded on the Lp.T/ spaces when 1 < p < 1. Using this, we give asimple proof for the analogous theorem for Fourier transforms in the next section.Recall that T WD Œ��; �� is the torus.

Definition 2.3.1 For an integrable function f 2 L1.T/ its kth Fourier coefficient isdefined by

bf .k/ D 1

2�

Z

T

f .x/e�{kx dx:

The formal trigonometric series

X

k2Zbf .k/e{kx .x 2 T/

is called the Fourier series of f .

Definition 2.3.2 For f 2 L1.T/ the nth partial sum sn f of the Fourier series of fand the nth Dirichlet kernel Dn are introduced by

sn f .x/ WDnX

kD�n

bf .k/e{kx .n 2 N/

and

Dn.u/ WDX

jkj�n

e{ku;

respectively.Clearly

sn f .x/ D 1

2�

Z

T

f .x � u/Dn.u/ du .n 2 N/:

Using some simple trigonometric identities, we obtain

Dn.u/ D 1C 2

nX

kD1cos.ku/

D 1

sin.u=2/

sin.u=2/C 2

nX

kD1cos.ku/ sin.u=2/

�

D 1

sin.u=2/

sin.u=2/CnX

kD1

sin..k C 1=2/u/� sin..k � 1=2/u/��

D sin..n C 1=2/u/

sin.u=2/:

It is easy to see that jDnj � Cn. The L1.T/-norms of Dn are not uniformly bounded,more exactly kDnk1 � log n.

It is a basic question as to whether the function f can be reconstructed from thepartial sums of its Fourier series. It can be found in most books about trigonometricFourier series (e.g. Zygmund [400], Bary [16], Torchinsky [330] or Grafakos [152])and is due to Riesz [271], that the partial sums converge to f in the Lp.T/-norm if1 < p < 1.

Theorem 2.3.3 If f 2 Lp.T/ for some 1 < p < 1, then

ksn f kp � Cp k f kp .n 2 N/ (2.3.1)

and

limn!1 sn f D f in the Lp.T/-norm: (2.3.2)

For the proof we need some other definitions and results. We follow the proof ofGrafakos [152].

Definition 2.3.4 For some n 2 N, the function

nX

kD�n

cke{kx .x 2 R/

is said to be a trigonometric polynomial.It is a well-known result that the trigonometric polynomials are dense in Lp.T/

for any 1 � p < 1.

Definition 2.3.5 For a trigonometric polynomial f define the conjugate functionefby

ef .x/ WD �{X

k2Zsign .k/bf .k/e{kx:

Also define the Riesz projections PC and P� by

PCf .x/ �1X

kD1bf .k/e{kx

and

P�f .x/ ��1X

kD�1bf .k/e{kx

Observe that f D PCf C P�f Cbf .0/ andef D �{PCf C {P�f .

Theorem 2.3.6 If 1 � p � 1, then (2.3.1) is equivalent to one of the nextinequalities

PCf

p � Cp k f kp (2.3.3)

and

ef

p� Cp k f kp ; (2.3.4)

where f is an arbitrary trigonometric polynomial.

Proof Since

PCf D 1

2. f C {ef /� 1

2bf .0/;

(2.3.3) is equivalent to (2.3.4). It is easy to see that

nX

kD�n

bf .k/e{kx D e�{nx2nX

kD04. f .�/e{n.�//.k/e{kx:

This implies that the norm of sn W Lp.T/ ! Lp.T/ is equal to the norm of PCn W

Lp.T/ ! Lp.T/, where

PCn g.x/ D

2nX

kD0bg.k/e{kx:

If (2.3.1) holds, then PCn is uniformly bounded on Lp.T/, hence (2.3.3) holds as

well.On the other hand, suppose that (2.3.3) holds and so PC can be extended to

Lp.R/. Then

PCn f D

1X

kD0bf .k/e{kx �

1X

kD2nC1bf .k/e{kx

D1X

kD0bf .k/e{kx � e{.2nC1/x

1X

kD0bf .k C 2n C 1/e{kx

D PCf .x/� e{.2nC1/xPC.e�{.2nC1/.�/f / �bf .0/.1� e{.2nC1/x/

for all trigonometric polynomials. By density this yields that

PCn f

p� �

2

PCC 2

� k f kp

for all f 2 Lp.R/ and n 2 N. Consequently, sn is uniformly bounded on Lp.T/,which is exactly (2.3.1). �

Now we show thatef is bounded on Lp.T/ .1 < p < 1/ (see Riesz [272, 273]).

Theorem 2.3.7 If 1 < p < 1, then

ef

p � Cp k f kp . f 2 Lp.T//:

Proof First suppose that f is a real trigonometric polynomial and bf .0/ D 0. It iseasy to see thatef is also real-valued and f C {ef contains only positive frequencies.

SinceR

Te{kx dx D 0 .k ¤ 0/, we have

Z

T

�

f .x/C {ef .x/�2k

dx D 0;

where k is a positive natural number. Taking the real part of the integral and usingthat f andef are real-valued, we obtain

kX

jD0.�1/k�j

2k

2j

!

Z

T

f .x/2 jef .x/2k�2j dx D 0:

This and Hölder’s inequality imply that

ef

2k

2k�

kX

jD1

2k

2j

!

Z

T

f .x/2 jef .x/2k�2j dx

�kX

jD1

2k

2j

!

k f k2j2k

ef

2k�2j

2k:

Let R D

ef

2k= k f k2k and divide by k f k2k to obtain

R2k �kX

jD1

2k

2j

!

R2k�2j � 0:

Then R is smaller than the largest root in absolute value of the polynomial on theleft-hand side, say R � C2k, in other words

ef

p � Cp k f kp for p D 2k: (2.3.5)

If bf .0/ ¤ 0, then apply this inequality to f �bf .0/. Since jbf .0/j � k f kp, we getthe preceding inequality with 2Cp. Every general trigonometric polynomial can bewritten as the sum of two real-valued trigonometric polynomials. Therefore (2.3.5)holds for every trigonometric polynomials and by density for all f 2 Lp.T/, p D2k. By interpolation (see Theorem 1.7.8 and Corollary 1.7.13), (2.3.5) holds for all2 � p < 1. Finally, observe that the adjoint operator of f 7!ef is f 7! �ef , whichimplies by duality that (2.3.5) holds also for 1 < p � 2. �Proof of Theorem 2.3.3 Inequality (2.3.1) follows from Theorems 2.3.6 and 2.3.7.The convergence (2.3.2) is clearly valid for trigonometric polynomials and so theconvergence follows for all f 2 Lp.T/ .1 < p < 1/ by density. �

One of the deepest results in harmonic analysis is Carleson’s theorem, thatthe partial sums of the Fourier series converge almost everywhere to f 2 Lp.T/

.1 < p � 1/. Since the proof can be found in many papers and books (see e.g.Carleson [52], Hunt [187], Arias de Reyna [8], Grafakos [152], Muscalu and Schlag[253], Lacey [207] or Demeter [88]), we present the result without proof.

Theorem 2.3.8 If f 2 Lp.T/ for some 1 < p < 1, then

supn2N

jsn f j

p

� Cp k f kp (2.3.6)

and if 1 < p � 1, then

limn!1 sn f D f a.e.

The inequality of Theorem 2.3.8 does not hold if p D 1 or p D 1, and thealmost everywhere convergence does not hold if p D 1. Du Bois Reymond provedthe existence of a continuous function f 2 C.T/ and a point x0 2 T such that thepartial sums sn f .x0/ diverge as n ! 1. Kolmogorov gave an integrable functionf 2 L1.T/, whose Fourier series diverges almost everywhere or even everywhere(see Kolmogorov [201, 202], Zygmund [400] or Grafakos [152]).

2.4 Convergence of the Inverse Fourier Transform

We turn back to the Fourier transforms. We have seen in the preceding two sectionsthat the inverse Fourier transform ofbf is f if f 2 L2.R/ or even if f 2 S0.R/. If inadditionbf 2 L1.R/, then we can write this in the form

f .x/ D 1p2�

Z

R

bf .t/ e{tx dt

for almost every x 2 R.

Corollary 2.4.1 If f 2 Lp.R/ for some 1 � p � 2 andbf 2 L1.R/, then for almostevery x 2 R,

f .x/ D 1p2�

Z

R

bf .t/ e{tx dt:

In casebf 62 L1.R/, the right-hand side does not make sense in general. Integratingon the right-hand side over the set fjtj � Tg, we obtain the definition of the Dirichletintegrals.

2.4 Convergence of the Inverse Fourier Transform 91

Definition 2.4.2 The Tth Dirichlet integral of the function f 2 Lp.R/ .1 � p � 2/

is given by

sT f .x/ WD 1p2�

Z T

�T

bf .t/ e{tx dt .x 2 R;T > 0/ :

Note that by Hausdorff-Young theorem (Theorem 2.1.13), bf 2 Lp0.R/ and sobf 2 Lp0.�T;T/ � L1.�T;T/, where 1=pC1=p0 D 1 and 1 � p � 2. In other words,

jsT f .x/j � C

bf

p0

T1=p � CT1=p k f kp : (2.4.1)

Thus sT f is well defined. If in addition f 2 L1.R/, then by Fubini’s theorem(Fig. 2.1),

sT f .x/ D 1

2�

Z T

�T

Z

R

f .y/ e�{yt dy e{tx dt DZ

R

f .y/DT.x � y/ dy: (2.4.2)

Definition 2.4.3 The Tth Dirichlet kernel is given by

DT.x/ WD 1

2�

Z T

�Te{xt dt .x 2 R;T > 0/ :

−3 −2 −1 0 1 2 3

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 2.1 Dirichlet kernel DT for T D 5

Proposition 2.4.4 We have jDT.x/j � T=� , jDT.x/j � 1=.�jxj/ and so DT 2Lp.R/ for any 1 0, x ¤ 0.

Proof The first and the last statements are trivial. The equality

DT.x/ D sin Tx

�x

finishes the proof. �Hölder’s inequality

ˇ

ˇ

ˇ

ˇ

Z

R

f .y/DT.x � y/ dy

ˇ

ˇ

ˇ

ˇ

� k f kp kDTkp0 (2.4.3)

implies that the right-hand side of (2.4.2) is defined for all f 2 Lp.R/ .1 � p < 1/.Using (2.4.1), (2.4.3) and the density argument, we conclude that (2.4.2) holds forall f 2 Lp.R/ .1 � p � 2/.

Definition 2.4.5 We extend the Tth Dirichlet integral to the functions f 2Lp.R/ .1 � p < 1/ by

sT f .x/ WDZ

R

f .y/DT.x � y/ dy .x 2 R;T > 0/ :

The analogue of Theorem 2.3.3 reads as follows. In the next two theorems wefollow the proofs of Grafakos [152].

Theorem 2.4.6 If f 2 Lp.R/ for some 1 0/ (2.4.4)

and

limT!1 sT f D f in the Lp.R/-norm: (2.4.5)

Proof We trace back the theorem to Theorem 2.3.3. Suppose that f and g arecontinuous functions with compact support. Then

FR.x/ WD f .Rx/ and GR.x/ WD g.Rx/

are supported in Œ��; �� when R is large enough, say R � R0. Observe that the kthFourier coefficient of FR is .2�/�1=2R�1

bf .k=R/ and of GR is .2�/�1=2R�1bg.k=R/.


Thenˇ

ˇ

ˇ

ˇ

ˇ

1

R

X

k2Z1Œ�T;T�.k=R/bf .k=R/bg.k=R/

ˇ

ˇ

ˇ

ˇ

ˇ

(2.4.6)

D .2�/1=2

ˇ

ˇ

ˇ

ˇ

ˇ

RX

k2Z1Œ�T;T�.k=R/cFR.k/cGR.k/

ˇ

ˇ

ˇ

ˇ

ˇ

D .2�/1=2

ˇ

ˇ

ˇ

ˇ

ˇ

RZ

T

X

k2Z1Œ�T;T�.k=R/cFR.k/e

{kx

!

GR.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

D .2�/1=2ˇ

ˇ

ˇ

ˇ

RZ

T

sbTRcFR.x/GR.x/ dx

ˇ

ˇ

ˇ

ˇ

:

By Hölder’s inequality and Theorem 2.3.3, this can be estimated by

.2�/1=2R

sbTRcFR

Lp.T/kGRkLp0 .T/ � CpR kFRkLp.T/

kGRkLp0 .T/

� Cp k f kLp.R/kgkLp0 .R/ ;

where 1=p C 1=p0 D 1. Sincebf andbg are continuous functions, by the definition ofthe Riemann integral, the left-hand side of (2.4.6) converges to

ˇ

ˇ

ˇ

ˇ

Z

R

1Œ�T;T�.t/bf .t/bg.t/ dt

ˇ

ˇ

ˇ

ˇ

as R ! 1. Now Theorem 2.1.8 implies that

ˇ

ˇ

ˇ

ˇ

Z

R

1Œ�T;T�bf�_.x/g.x/ dx

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

Z

R

1Œ�T;T�.t/bf .t/bg.t/ dt

ˇ

ˇ

ˇ

ˇ

� Cp k f kp kgkp0 :

Taking the supremum over all g’s with kgkp0 � 1, we conclude that

ksT f kp � Cp k f kp .T > 0/:

A usual density argument shows that (2.4.4) is valid for all f 2 Lp.R/.The convergence (2.4.5) follows obviously from Definition 2.4.2 for all f 2

L1.R/ the Fourier transforms of which have compact supports. By Corollary 2.6.6this set is dense in Lp.R/ and so the theorem follows by density. �

The inequality of Theorem 2.4.6 does not follow from (2.4.3) because DT is notuniformly bounded in Lp0.R/. Carleson’s theorem can be formulated for Fouriertransforms as follows.

Theorem 2.4.7 If f 2 Lp.R/ for some 1 0

jsT f j

p

� Cp k f kp

and

limT!1 sT f D f a.e.

Proof By Lemma 1.8.2 it is enough to prove that

sup0<T1<:::<Tn

ˇ

ˇsTj fˇ

ˇ

p

� Cp k f kp : (2.4.7)

The operator sup0<T1<:::<Tn

ˇ

ˇsTj

ˇ

ˇ can be viewed as an operator from Lp.R/ toLp.R; `1.F//, where F D fT1; : : : ;Tng. By duality, (2.4.7) is equivalent to

nX

jD1sTj fj

p0

� Cp

nX

jD1

ˇ

ˇ fjˇ

ˇ

p0

; (2.4.8)

where 1=p C 1=p0 D 1. Similarly, (2.3.6) is equivalent to

nX

jD1snj fj

Lp0 .T/

� Cp

nX

jD1

ˇ

ˇ fjˇ

ˇ

Lp0 .T/

: (2.4.9)

Let fj and g be continuous functions with compact support and

Fj;R.x/ WD fj.Rx/ and GR.x/ WD g.Rx/ . j D; 1; : : : n/

as in Theorem 2.4.6. Similar to that proof,

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1

R

X

k2Z

nX

jD11Œ�Tj;Tj�.k=R/bfj.k=R/bg.k=R/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

(2.4.10)

D .2�/1=2

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

RZ

T

0

@

nX

jD1

X

k2Z1Œ�Tj ;Tj�.k=R/bFj;R.k/e

{kx

1

AGR.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

D .2�/1=2

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

RZ

T

nX

jD1sbTjRcFj;R.x/GR.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2.5 Summability of Fourier transforms 95

� .2�/1=2R

nX

jD1sbTjRcFj;R

Lp0 .T/

kGRkLp.T/

� CpR

nX

jD1

ˇ

ˇFj;R

ˇ

ˇ

Lp0 .T/

kGRkLp.T/

� Cp

nX

jD1

ˇ

ˇ fjˇ

ˇ

Lp0 .R/

kgkLp.R/:

Here we have used (2.4.9). The left-hand side of (2.4.10) converges to

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R

nX

jD11Œ�Tj ;Tj�.t/bfj.t/bg.t/ dt

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R

nX

jD1

1Œ�Tj;Tj �bfj�_.x/g.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

as R ! 1. Taking the supremum over all g’s with kgkp � 1, we obtain (2.4.8).Using the density argument presented in Theorem 2.4.6, the convergence can beshown easily. �

2.5 Summability of One-Dimensional Fourier Transforms

Though Theorems 2.4.6 and 2.4.7 are not true for p D 1 and p D 1, with thehelp of some summability methods they can be generalized for these endpointcases. Obviously, summability means have better convergence properties than theoriginal Fourier series. Summability is intensively studied in the literature. We referat this time only to the books Stein and Weiss [311], Butzer and Nessel [46],Trigub and Belinsky [339], Grafakos [152] and Weisz [355] and the referencestherein.

The best known summability method is the Fejér method. In 1904 Fejér [116]investigated the arithmetic means of the partial sums, the so-called Fejér means andproved that if the left and right limits f .x � 0/ and f .x C 0/ exist at a point x, thenthe Fejér means converge to . f .x � 0/ C f .x C 0//=2. One year later Lebesgue[212] extended this theorem and obtained that every integrable function is Fejérsummable at each Lebesgue point, thus almost everywhere. The Riesz means aregeneralizations of the Fejér means. Riesz [271] proved that the Riesz means of afunction f 2 L1.T/ converge almost everywhere to f as n ! 1 (see also Zygmund[400, Vol. I, p.94]).

In this section we consider a very general summability method, the so-called�-summability. In the investigation of the �-summability of Fourier transforms we


will always assume that

� 2 L1.R/ \ C0.R/ and �.0/ D 1:

Suppose first that f 2 Lp.R/ for some 1 � p � 2. We modify the definition of theDirichlet integral and introduce the �-means as follows.

Definition 2.5.1 The Tth �-mean of the function f 2 Lp.R/ .1 � p � 2/ is givenby

��T f .x/ WD 1p2�

Z

R

��t

T

�

bf .t/e{xt dt: (2.5.1)

The integral is well defined becausebf 2 Lp0.R/ if f 2 Lp.R/ .1 � p � 2; 1=p C1=p0 D 1/ and � 2 L1.R/ \ C0.R/ implies � 2 Lp.R/ .1 � p � 1/. Moreover, byHölder’s inequality and Hausdorff-Young theorem,

ˇ

ˇ��T f .x/ˇ

ˇ � C

bf

p0

T1=p k�kp � CT1=p k f kp k�kp ; (2.5.2)

where 1=p C 1=p0 D 1 and 1 � p � 2. For � D 1Œ�1;1� we would get back theDirichlet integral sT f ; however, the characteristic function is not continuous. It iseasy to see that for an integrable function f , we can rewrite ��T f as

��T f .x/ DZ

R

f .x � t/K�T .t/ dt D f � K�

T .x/ .x 2 R;T 2 RC/: (2.5.3)

Definition 2.5.2 The Tth �-kernel is given by

K�T .x/ WD 1

2�

Z

R

��t

T

�

e{xt dt D 1p2�

Tb�.Tx/ .x 2 R;T 2 RC/:

Forb� 2 L1.R/, it follows from Definition 2.5.2 that

K�T

1D 1p

2�

b�

1.T 2 RC/:

Sinceb� 2 L1.R/\ L1.R/, we haveb� 2 Lp.R/ and K�T 2 Lp.R/ for all 1 � p � 1.

Again by Hölder’s inequality,

ˇ

ˇ

ˇ

ˇ

Z

R

f .x � t/K�T .t/ dt

ˇ

ˇ

ˇ

ˇ

� k f kp

K�T

p0:

This, (2.5.2) and the density argument imply that (2.5.3) holds for all f 2 Lp.R/ .1 �p � 2/.

2.5 Summability of Fourier transforms 97

Forb� 2 L1.R/, we can extend the definition of the �-means in the following way.

Definition 2.5.3 Ifb� 2 L1.R/, then we extend the Tth �-mean to all f 2 Lp.R/ .1 �p � 1/ or even to all f 2 W.L1; `1/.R/ by

��T f WD f � K�T .T 2 RC/:

Note that the Wiener amalgam space W.L1; `1/.R/ will be introduced in thenext section. Thus the �-means can be rewritten as

��T f .x/ D Tp2�

Z

R

f .x � t/b�.Tt/ dt: (2.5.4)

The most known summability method is the Fejér summation (see Fig. 2.2) when

�.t/ WD�

1 � jtj; if jtj � 1I0; if jtj > 1:

In this case the �-kernel is called Fejér kernel and

b�.x/ D 1p2�

�

sin x=2

x=2

�2

:

−3 −2 −1 0 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 2.2 Fejér kernel K�T for T D 5

For an even and differentiable function � , the same definition implies

K�T .x/ D 1

2�

Z

R

�

� jujT

�

e{xu du

D �12�T

Z

R

Z 1

juj� 0 t

T

�

dt e{xu du

D �12�T

Z 1

0

� 0 t

T

�

Z t

�te{xu du dt

D �1T

Z 1

0

� 0 t

T

�

Dt.x/ dt:

Hence

��T f .x/ D �1T

Z 1

0

� 0 t

T

�

st f .x/ dt:

Note that for the Fejér means, we obtain

��T f .x/ D 1

T

Z T

0

st f .x/ dt:

2.6 Norm Convergence of the Summability Means

We need some new function spaces which generalize the Lp.R/ spaces.

Definition 2.6.1 A measurable function f belongs to the Wiener amalgam spaceW.Lp; `q/.R/ .1 � p; q � 1/ if

k f kW.Lp;`q/WD

X

k2Zk f .� C k/kq

LpŒ0;1/

!1=q

< 1;

with the obvious modification for q D 1. If we replace the space LpŒ0; 1/ byLp;1Œ0; 1/, then we get the definition of W.Lp;1; `q/.R/. The closed subspace ofW.L1; `q/.R/ containing continuous functions is denoted by W.C; `q/.R/ .1 �q � 1/. W.Lp; c0/.R/ is defined analogously .1 � p � 1/. The spaceW.L1; `1/.R/ is called Wiener algebra.

It is easy to see that W.Lp; `p/.R/ D Lp.R/ and the following continuousembeddings hold:

W.Lp1 ; `q/.R/ W.Lp2 ; `q/.R/ . p1 � p2/

2.6 Norm Convergence of the Summability Means 99

and

W.Lp; `q1 /.R/ � W.Lp; `q2 /.R/ .q1 � q2/;

.1 � p1; p2; q1; q2 � 1/. Thus

W.L1; `1/.R/ � Lp.R/ � W.L1; `1/.R/ .1 � p � 1/:

Definition 2.6.2 A Banach space B consisting of Lebesgue measurable functionson R is called a homogeneous Banach space if

(i) for all f 2 B and x 2 R, Tx f 2 B and kTx f kB � k f kB,(ii) the function x 7! Tx f from R to B is continuous for all f 2 B,

(iii) the functions in B are uniformly locally integrable, i.e. for every compact setK � R there exists a constant CK such that

Z

Kj f j d� � CKk f kB . f 2 B/:

Note that if B is a homogeneous Banach space on R, then

B � W.L1; `1/.R/

(see Katznelson [196] or Shapiro [295]). It is easy to see that the spaces C0.R/,Lp.R/, W.Lp; `q/.R/, W.Lp; c0/.R/, W.C; `q/.R/ .1 � p; q < 1/, Lorentz spacesLp;q.R/ .1 < p < 1; 1 � q < 1/ and Hardy space H1.R/ are all homogeneousBanach spaces. Moreover, the space Cu.R/ of uniformly continuous boundedfunctions endowed with the supremum norm is also a homogeneous Banach space.

The next result holds also ifb� is not integrable.

Theorem 2.6.3 For all f 2 L2.R/,

limT!1 ��T f D f in the L2.R/-norm.

Proof Let f 2 L1.R/ the Fourier transform of which has compact support. ByCorollary 2.6.6, the set of these functions is dense in L2.R/. Then (2.5.3) holds.It is easy to see that the norm of the operator ��T W L2.R/ ! L2.R/ is

supk f k2�1

Tp2�

f �b�.T�/

2D 1p

2�sup

k f k2�1

f _�.�=T/

2

D 1p2�

supkgk2�1

kg �.�=T/k2

D 1p2�

k�k1 :


Thus the norms of ��T .T 2 RC/ are uniformly bounded. The Fourier inversionformula and (2.5.1) imply

��T f .x/ � f .x/ D 1p2�

Z

R

��t

T

�

� 1�

bf .t/e{xt dt

D F�1

��

T

�

� 1�

bf .�/�

.x/:

Thus

��T f � f

2D

��

T

�

� 1�

bf .�/

2! 0

as T ! 1, because � is continuous. The density theorem completes the proof. �If in additionb� is integrable, we can prove the following more general result.

Theorem 2.6.4 Assume that B is a homogeneous Banach space on R. If � 2 L1.R/andb� 2 L1.R/, then

��T f

B� C k f kB . f 2 B;T 2 RC/

and

limT!1 ��T f D f in the B-norm for all f 2 B:

Proof Using

1p2�

Z

R

b�.t/ dt D �.0/ D 1

and (2.5.4), we conclude

��T f .x/� f .x/ D 1p2�

Z

R

f

x � t

T

�

� f .x/�

b�.t/ dt

D 1p2�

Z

R

T tT

f .x/ � f .x/�

b�.t/ dt

and

��T f � f

B � 1p2�

Z

R

T tT

f � f

B

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ dt:

The theorem follows from (i) and (ii) of the definition of the homogeneous Banachspaces and from the Lebesgue dominated convergence theorem. �

2.7 Almost everywhere convergence 101

Corollary 2.6.5 If f is a uniformly continuous and bounded function, � 2 L1.R/andb� 2 L1.R/, then

limT!1 ��T f D f uniformly.

Corollary 2.6.6 The set of the functions f 2 L1.R/ the Fourier transforms of whichhave compact supports is dense in Lp.R/ for all 1 � p < 1.

Proof We know that L1.R/ \ C0.R/ is dense in Lp.R/ for all 1 � p < 1. Itis enough to prove that all functions f 2 L1.R/ \ C0.R/ can be approximatedby integrable functions the Fourier transforms of which have compact supports.Suppose that � has compact support, say

�.t/ WD�

1 � jtj; if jtj � 1I0; if jtj > 1

is the Fejér summation. It is clear by Theorem 2.6.4 that ��T f 2 Lp.R/ for all 1 �p < 1 and ��T f ! f in the Lp.R/-norm as T ! 1. We have to show yet that theFourier transform of ��T f has compact support. Since

��

T

�

bf .�/ 2 L1.R/\ L2.R/;

Definition 2.5.2 and (2.5.3) imply that

b��T f .t/ D ��t

T

�

bf .t/:

From this it follows that the support of b��T f is Œ�T;T� �

2.7 Almost Everywhere Convergence of the SummabilityMeans

In this section we will prove that the �-means converge almost everywhere to f ifb�is in a Herz space. So we introduce first the Herz spaces.

Definition 2.7.1 A function f 2 Llocq .R/ is in the Herz space Eq.R/ if

k f kEqWD

f1I.0;1/

qC

1X

kD12k.1�1=q/ k f1Pk kq < 1;

where Pk WD I.0; 2k/ n I.0; 2k�1/, .k 2 Z/ and 1 � q � 1.

We modify slightly this definition and introduce the homogeneous version of thisspace.

Definition 2.7.2 The homogeneous Herz space PEq.R/ contains all functions f 2Lloc

q .R/ for which

k f k PEqWD

1X

kD�12k.1�1=q/ k f1Pkkq < 1: (2.7.1)

These spaces are special cases of the Herz spaces [178]. Herz spaces havedifferent but equivalent norms (see Garcia-Cuerva and Herrero [125], Feichtinger[111]). We have

Eq.R/ � PEq.R/; Eq.R/ � Lq.R/; k f k PEq� Cq k f kEq

; k f kq � Cq k f kEq:

Indeed,

0X

kD�12k.1�1=q/ k f1Pkkq �

0X

kD�12k.1�1=q/

f1I.0;1/

q� Cq k f kEq

and the second inequality is trivial. Moreover, it is easy to see that

Eq.R/ D Lq.R/ \ PEq.R/ and k f kEq � k f kq C k f k PEq

with equivalent norms. Since b� 2 L1.R/ \ C0.R/, b� 2 PEq.R/ if and only if b� 2Eq.R/, but kb�k PEq

� Cqkb�kEq , .1 � q � 1/. Furthermore, we have the followingembeddings:

L1.R/ D PE1.R/ PEq.R/ PEq0.R/ PE1.R/ 1 < q < q0 < 1:

We show that f 2 PE1.R/ if and only if f has an integrable decreasing majorantfunction.

Theorem 2.7.3 Let �.x/ WD supjtj�jxj j f .t/j. Then f 2 PE1.R/ if and only if � 2L1.R/ and

C�1k�k1 � k f k PE1

� Ck�k1:

Proof If � 2 L1.R/, then

k f k PE1

� k�k PE1

D1X

kD�12kk�1Pk k1 D

1X

kD�12k�.2k�1/ � Ck�k1:


For the converse denote by

ak WD supPk

j f j and �0 WD1X

kD�1ak1Pk :

Let

�.x/ WD supjtj�jxj

�0.t/ .x 2 R/:

Since f 2 PE1.R/ implies limk!1 ak D 0, we conclude that there exists anincreasing sequence .nk/k2Z of integers such that .ank/k2Z is decreasing and � canbe written in the form

� D1X

kD�1ank1I.0;2nk /nI.0;2nk�1 /:

Thus

k�k1 � k�k1 D1X

kD�1ank

Z

I.0;2nk /nI.0;2nk�1 /

d� D C1X

kD�1.2nk � 2nk�1 / ank :

By Abel rearrangement

k�k1 � C1X

kD�12nk�1 .ank�1 � ank/ � Ck f k PE1

;

which proves the theorem. �To prove pointwise convergence of the �-means we have to investigate the

maximal operator.

Definition 2.7.4 The maximal operator �� f is defined by

�� f WD supT>0

ˇ

ˇ��T fˇ

ˇ :

Ifb� 2 L1.R/, then (2.5.4) implies

�� f

1 �

b�

1k f k1 . f 2 L1.R//: (2.7.2)

As we will see later, this inequality is valid for Lp.R/-norms and for all f 2 Lp.R/

as well.

Theorem 2.7.5 Let � 2 L1.R/, 1 � p < 1 and 1=p C1=q D 1. Ifb� 2 PEq.R/, thenfor all f 2 Lloc

p .R/,

�� f .x/ � C

b�

PEq

Mp f .x/:

Proof By (2.5.4),

ˇ

ˇ��T f .x/ˇ

ˇ D Tp2�

ˇ

ˇ

ˇ

ˇ

Z

R

f .x � t/b�.Tt/ dt

ˇ

ˇ

ˇ

ˇ

� Tp2�

1X

kD�1

Z

Pk.T/j f .x � t/j

ˇ

ˇ

ˇ

b�.Tt/ˇ

ˇ

ˇ dt;

where

Pk.T/ WD I.0; 2k=T/ n I.0; 2k�1=T/ .k 2 Z/: (2.7.3)

By Hölder’s inequality,ˇ

ˇ��T f .x/ˇ

ˇ

� Tp2�

1X

kD�1

�Z

Pk.T/

ˇ

ˇ

ˇ

b�.Tt/ˇ

ˇ

ˇ

qdt

�1=q �Z

Pk.T/j f .x � t/jp dt

�1=p

D T1�1=q

p2�

1X

kD�1

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q �Z

Pk.T/j f .x � t/jp dt

�1=p

: (2.7.4)

If we define

G.u/ WD�Z u

�uj f .x � t/jp dt

�1=p

.u > 0/;

then

Gp.u/

2u� Mp

p f .x/ .u > 0/:

Hence

ˇ

ˇ��T f .x/ˇ

ˇ � CT1=p1X

kD�1

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q

G

�

2k

T

�

� C1X

kD�12k=p

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q

Mp f .x/

� C

b�

PEq

Mp f .x/

and the theorem is shown. �

The next result follows easily from Corollary 1.2.5.

Theorem 2.7.6 Let � 2 L1.R/, 1 � p � 1 and 1=p C1=q D 1. Ifb� 2 PEq.R/, then

�� f

p;1 D sup�>0

��.�� f > �/1=p � Cp

b�

PEq

k f kp

for all f 2 Lp.R/. Moreover, for every p < r � 1,

�� f

r� Cr

b�

PEq

k f kr . f 2 Lr.R//:

Corollary 2.7.7 If � 2 L1.R/, 1 � p � 1, 1=p C 1=q D 1 andb� 2 PEq.R/, then

limT!1 ��T f D f a.e.

when f 2 Lr.R/ for p � r < 1 or f 2 C0.R/.

Proof For f 2 Cc.R/, we obtain the convergence from

ˇ

ˇ��T f .x/ � f .x/ˇ

ˇ � 1p2�

Z

R

ˇ

ˇ

ˇ f

x � t

T

�

� f .x/ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ dt

and from Lebesgue dominated convergence theorem. Since Cc.R/ is dense inLp.R/ .1 � p < 1/ and in C0.R/, the corollary follows from Theorems 2.7.6and 1.2.6. �

Note that PEq.R/ PEq0.R/ whenever q < q0. If b� is in a smaller space (say inPE1.R/) then we get convergence for a wider class of functions (namely for f 2Lr.R/, 1 � r � 1).

We will generalize the last theorem and corollary for the larger spaceW.L1; `1/.R/. First we have to generalize the Hardy-Littlewood maximal function.

Definition 2.7.8 For f 2 Llocp .R/ and 1 � p < 1 the local Hardy-Littlewood

maximal function is given by

mpf .x/ WD sup0<r�1

�

1

jI.x; r/jZ

I.x;r/j f jp d�

�1=p

.x 2 R/:

It is easy to see that Corollary 1.2.5 implies

kmpf kW.Lp;1 ;`s/ � Cpk f kW.Lp ;`s/ . f 2 W.Lp; `s/.R//

and

kmpf kW.Lr ;`s/ � Crk f kW.Lr ;`s/ . f 2 W.Lr; `s/.R//

for all p < r � 1 and 1 � s � 1. Recall that

k f kW.Lp;1 ;`1/ D supk2Z

sup�>0

� �.j f j > �; Œk; k C 1//1=p;

where

�.j f j > �; Œk; k C 1// WD �.1Œk;kC1/j f j > �/:

Corollary 2.7.9 If 1 � p < 1, then

Mp f

W.Lp;1;`1/� Cp k f kW.Lp ;`1/

for all f 2 W.Lp; `1/.R/. Moreover, for every p < r � 1,

Mp f

W.Lr ;`1/� Cr k f kW.Lr ;`1/ . f 2 W.Lr; `1/.R//:

Proof The corollary follows from the inequality

Mp f � Cmp f C Cpk f kW.Lp ;`1/ .1 � p � 1/:

and from Corollary 1.2.5. �This and Theorem 2.7.5 imply

Theorem 2.7.10 Let � 2 L1.R/, 1 � p � 1 and 1=p C 1=q D 1. Ifb� 2 PEq.R/,then

�� f

W.Lp;1;`1/� Cp

b�

PEq

k f kW.Lp;`1/

for all f 2 W.Lp; `1/.R/. Moreover, for every p < r � 1,

�� f

W.Lr ;`1/� Cr

b�

PEq

k f kW.Lr ;`1/ . f 2 W.Lr; `1/.R//:

Corollary 2.7.11 If � 2 L1.R/, 1 � p < 1, 1=p C 1=q D 1 andb� 2 PEq.R/, thenfor all f 2 W.Lp; c0/.R/,


Proof The density of Cc.R/ in W.Lp; c0/.R/ and a slightly modified version ofTheorem 1.2.6 imply the corollary. �

Note that W.Lp; c0/.R/ contains all W.Lr; c0/.R/ and Lr.R/ spaces for p �r � 1. The converse of Theorem 2.7.5 holds also. More exactly, if �� f can be

estimated pointwise by Mp f , thenb� 2 PEq.R/. Before proving this theorem, we needthe following definition.

Definition 2.7.12 We define the space PDp.R/ .1 � p < 1/ by the norm

k f k PDpWD sup

r>0

�

1

r

Z r

�rj f jp d�

�1=p

: (2.7.5)

Lemma 2.7.13 The norm

k f k� D supk2Z

2�k=p k f1Pk kp (2.7.6)

is an equivalent norm on PDp.R/, where Pk D I.0; 2k/ n I.0; 2k�1/, .k 2 Z/ and1 � p < 1.

Proof Obviously,

2�k=p k f1Pkkp �

1

2k

Z 2k

�2kj f jp d�

!1=p

� k f k PDp:

On the other hand, suppose that 2N � r < 2NC1. Then

1

r

Z r

�rj f jp d� � 2�N

Z 2NC1

�2NC1

j f jp d�

D 2�NNC1X

kD�1

Z

Pk

j f jp d�

� 2�NNC1X

kD�12k k f kp

� � C k f kp� ;

which shows (2.7.6). �Theorem 2.7.14 Let � 2 L1.R/,b� 2 L1.R/, 1 � p < 1 and 1=p C 1=q D 1. If

�� f .x/ � CMp f .x/ (2.7.7)

for all f 2 Lp.R/ and x 2 R, thenb� 2 PEq.R/.

Proof It is easy to see by (2.7.6) that

supk f k

PDp�1

ˇ

ˇ

ˇ

ˇ

Z

R

f .t/g.t/ dt

ˇ

ˇ

ˇ

ˇ

D Ckgk PEq.1 � p < 1/ (2.7.8)

and there exists a function f 2 PDp.R/ with k f k PDp� 1 such that

Ckgk PEq

2�ˇ

ˇ

ˇ

ˇ

Z

R

f .t/g.t/ dt

ˇ

ˇ

ˇ

ˇ

D supn2N

ˇ

ˇ

ˇ

ˇ

Z

R

fn.t/g.t/ dt

ˇ

ˇ

ˇ

ˇ

;

where fn WD f1I.0;2n/. This holds even if g 62 PEq.R/ and kgk PEqD 1. By (2.7.7),

ˇ

ˇ

ˇ

ˇ

Z

R

f .�t/b�.t/ dt

ˇ

ˇ

ˇ

ˇ

� CMp f .0/ . f 2 Lp.R//:

Since fn 2 Lp.R/, taking g Db� and the function f 2 PDp.R/ given above, we get that

b�

PEq

2� C sup

n2NMp fn.0/ � CMp f .0/ � Ck f k PDp

� C;

which shows thatb� 2 PEq.R/. �Note that the results of this section were proved in Feichtinger and Weisz

[112, 113].

2.8 Boundedness of the Maximal Operator

Using other conditions on � , we give another proof for the almost everywhereconvergence of ��T f . We have seen in (2.7.2) that the maximal operator �� is

bounded from L1.R/ to L1.R/ if b� 2 L1.R/. Under the stronger conditionb� 2 PE1.R/, Theorem 2.7.6 says that �� is bounded from Lp.R/ to Lp.R/

for all 1 < p � 1. This result does not hold for p D 1. However, usingHardy spaces, we can extend the theorem to p � 1. More exactly, we willshow that �� is bounded from Hp.R/ to Lp.R/ for all p0 < p � 1, where p0is depending on � . The weak type .1; 1/ inequality of �� follows from this byinterpolation.

In this section we do not use the conditionb� 2 PE1.R/. Instead, we assume thatb�is .N C1/-times differentiable for some N 2 N and there exists N C1 < ˇ � N C2

such thatˇ

ˇ

ˇ

ˇ

b��.k/

.x/

ˇ

ˇ

ˇ

ˇ

� Cjxj�ˇ .x ¤ 0/; (2.8.1)

whenever k D N and k D N C 1.

2.8 Boundedness of the Maximal Operator 109

Theorem 2.8.1 Assume thatb� 2 L1.R/ satisfies (2.8.1). If 1=ˇ 0

��.�� f > �/ˇ � C k f kH1=ˇ : (2.8.3)

Proof Let a be an arbitrary p-atom with support I and radius and let

2�K�1 < � 2�K .K 2 Z/:

We may suppose that the centre of I is zero, i.e.

Œ�2�K�1; 2�K�1� � I � Œ�2�K ; 2�K �:

Since �� is bounded from L1.R/ to L1.R/, by Theorems 1.8.3 and 1.8.5, it isenough to show that

Z

Rn4I

ˇ

ˇ�� a.x/ˇ

ˇ

pdx � Cp (2.8.4)

for all 1=ˇ 0.Obviously,

Z

Rn4I

ˇ

ˇ�� a.x/ˇ

ˇ

pdx

�1X

iD2

Z .iC1/2�K

i2�Ksup

T�2K

ˇ

ˇ��T a.x/ˇ

ˇ

pdx C

1X

iD2

Z .iC1/2�K

i2�Ksup

T<2K

ˇ

ˇ��T a.x/ˇ

ˇ

pdx

DW .A/C .B/:

As we mentioned before Definition 1.6.12, we can suppose that N � N. p/. We useTaylor’s formula

g.t/ DN�1X

kD0

g.k/.0/

kŠtk C g.N/.�t/

NŠtN

for g.t/ Db�.T.x � t//, where 0 < � < 1. Then

g.k/.t/ D .�1/kTk

b��.k/

.T.x � t//:


By the definition of the atom and by Taylor’s formula

��T a.x/ D Tp2�

Z

Ia.t/b�.T.x � t// dt

D Tp2�

Z

Ia.t/

b�.T.x � t// �N�1X

kD0

g.k/.0/

kŠtk

!

dt

D T

NŠp2�

Z

Ia.t/.�1/NTN

b��.N/

.T.x � �t//tN dt:

Then by (2.8.1),

ˇ

ˇ��T a.x/ˇ

ˇ � CTNC1Z

Ija.t/j jT.x � �t/j�ˇ jtjN dt:

If t 2 I and x 2 �i2�K ; .i C 1/2�K�

for some i � 2, then

jx � �tj � jxj � jtj � i2�K � 2�K D .i � 1/2�K:

In case of T � 2K , this implies that

ˇ

ˇ��T a.x/ˇ

ˇ � CTNC1�ˇ.i � 1/�ˇ2�K.N�ˇ/Z

Ija.t/j dt

� CTNC1�ˇ.i � 1/�ˇ2�K.N�ˇ/2K=p�K

� C.i � 1/�ˇ2K=p: (2.8.5)

Then

.A/ D1X

iD2

Z .iC1/2�K

i2�Ksup

T�2K

ˇ

ˇ��T a.x/ˇ

ˇ

pdx

� Cp

1X

iD2

Z .iC1/2�K

i2�K.i � 1/�ˇp2K dx

D Cp

1X

iD2.i � 1/�ˇp;

which is a convergent series since p > 1=ˇ. Obviously, this holds also for N D 0.Similarly, using Taylor’s formula for N C 1 instead of N, we have

��T a.x/ D Tp2�

Z

Ia.t/.�1/NC1TNC1

b��.NC1/

.T.x � �t//tNC1 dt

2.8 Boundedness of the Maximal Operator 111

and

ˇ

ˇ��T a.x/ˇ

ˇ � CTNC2Z

Ija.t/j jT.x � �t/j�ˇ jtjNC1 dt

� CTNC2�ˇ.i � 1/�ˇ2�K.NC1�ˇ/Z

Ija.t/j dt

� CTNC2�ˇ.i � 1/�ˇ2�K.NC1�ˇ/2Kd=p�K (2.8.6)

� C.i � 1/�ˇ2K=p

if T < 2K . As above

.B/ D1X

iD2

Z .iC1/2�K

i2�Ksup

T<2K

ˇ

ˇ��T a.x/ˇ

ˇ

pdx � Cp

1X

iD2.i � 1/�ˇp < 1:

This proves (2.8.4) as well as (2.8.2) for 1=ˇ �C�12�K=p�

;

where p D 1=ˇ. Observe by (2.8.5) that

�p �

(

supT�2K

ˇ

ˇ��T a.x/ˇ

ˇ > �

)

\ fR n 4Ig!

� �pX

i2E�

2�K :

If k is the largest integer, for which .i � 1/�ˇ > �C�12�K=p, then

�p �

(

supT�2K

ˇ

ˇ��T a.x/ˇ

ˇ > �

)

\ fR n 4Ig!

� �p2�Kk � C:

We can estimate supT<2K j��T a.x/j similarly and then (2.8.3) follows from Theo-rem 1.8.7. �

If in (2.8.1) ˇ D N C2, then we do not need the inequality for the Nth derivative.

Theorem 2.8.2 Let b� 2 L1.R/ be .N C 1/-times differentiable for some N 2 N.Assume that

ˇ

ˇ

ˇ

ˇ

b��.NC1/

.x/

ˇ

ˇ

ˇ

ˇ

� Cjxj�.NC2/ .x ¤ 0/: (2.8.7)

Then (2.8.2) holds for all 1=.N C 2/ < p < 1 and

�� f

1=.NC2/;1 � C1=.NC2/ k f kH1=.NC2/. f 2 H1=.NC2/.R//:

Proof Observe that the right-hand side in (2.8.6) is independent of T. Then

Z

Rn4I

ˇ

ˇ��a.x/ˇ

ˇ

pdx D

1X

iD2

Z .iC1/2�K

i2�K

ˇ

ˇ��a.x/ˇ

ˇ

pdx

� Cp

1X

iD2

Z .iC1/2�K

i2�K.i � 1/�.NC2/p2K dx < 1

and this completes the proof. �Recall that Hp.R/ � Lp.R/ for 1 < p � 1 and so (2.8.2) yields

�� f

p� Cp k f kp . f 2 Lp.R/; 1 < p � 1/:

If p is smaller than or equal to 1=ˇ, then this theorem is not true in general (seeOswald [264], Stein et al. [312], Weisz [352]). For the Fejér summation ˇ D 2, so,in this case, �� is bounded from Hp.R/ to Lp.R/ if 1=2 < p � 1. Furthermore, thefollowing theorem also holds.

Theorem 2.8.3 For the Fejér summability, the operator �� is not bounded fromHp.R/ to Lp.R/ if p � 1=2.

Of course, (2.8.3) cannot be true for p < 1=ˇ, i.e. �� is not bounded from Hp.R/

to the weak Lp;1.R/ space for p < 1=ˇ. If the operator was bounded, then byinterpolation (2.8.3) would hold for p D 1=ˇ, which contradicts Theorem 2.8.3.

Theorem 2.8.4 Under the conditions of Theorem 2.8.1 or 2.8.2, we have

�� f

1;1 D sup�>0

��.�� f > �/ � C k f k1 . f 2 L1.R//:

As we have seen in the previous section, this weak type inequality ensures thealmost everywhere convergence of ��T f .

2.9 Convergence at Lebesgue Points

Under some conditions on � , we can characterize the set of almost everywhereconvergence. The well-known theorem of Lebesgue [212] says that, for the Fejérmeans and for all f 2 L1.T/,

limn!1 �n f .x/ D f .x/ (2.9.1)

2.9 Convergence at Lebesgue Points 113

at each Lebesgue point of f . In this section, we generalize this result to othersummability methods and to the Wiener amalgam spaces (see Feichtinger and Weisz[113]).

First of all, we introduce the concept of Lebesgue points. Lebesgue differentia-tion theorem (Corollary 1.2.8) says that

limh!0

1

2h

Z h

�hf .x � u/ du D f .x/

for almost every x 2 R, where f 2 Lloc1 .R/. Thus

limh!0

1

2h

Z h

�h. f .x � u/� f .x// du D 0 a.e. x 2 R;

which is equivalent to

limh!0

1

2h

ˇ

ˇ

ˇ

ˇ

Z h

�h. f .x � u/� f .x// du

ˇ

ˇ

ˇ

ˇ

D 0 a.e. x 2 R:

The definition of the Lebesgue point is a stronger condition; however, almost everypoint will be a Lebesgue point of a function.

Definition 2.9.1 A point x 2 R is called a p-Lebesgue point of f if

limh!0

�

1

2h

Z h

�hj f .x � u/� f .x/jp du

�1=p

D 0 .1 � p < 1/ :

One can show that if p < r and x is an r-Lebesgue point of f , then it is also ap-Lebesgue point. Indeed, by Hölder’s inequality,

�

1

2h

Z h

�hj f .x � u/� f .x/jp du

�1=p

��

1

2h

Z h

�hj f .x � u/� f .x/jr du

�1=r

:

Theorem 2.9.2 Almost every point x 2 R is a p-Lebesgue point of f 2W.Lp; `1/.R/ .1 � p < 1/.

Proof For all rational numbers q let

Gq WD�

x 2 R W limh!0

1

2h

Z h

�hj f .x � t/ � qjp dt D j f .x/ � qjp

�

:

Applying Lebesgue’s theorem (see Corollary 1.2.8) to the function j f .�/� qjp, wecan see that Bq WD R n Gq is of Lebesgue measure 0. If f 2 W.Lp; `1/.R/, then f is

almost everywhere finite. Set N WD fx 2 R W j f .x/j D 1g. Then the set

B WD N[

0

@

[

q2QBq

1

A

has Lebesgue measure 0. We show that the points of G WD R n B are Lebesguepoints. Let � > 0 and x 2 G be arbitrary. Choose q 2 Q such that

j f .x/ � qj < �

2:

Then

�

1

2h

Z h

�hj f .x � t/ � f .x/jp dt

�1=p

��

1

2h

Z h

�hj f .x � t/ � qjp dt

�1=p

C�

1

2h

Z h

�hjq � f .x/jp dt

�1=p

D�

1

2h

Z h

�hj f .x � t/ � qjp dt

�1=p

C jq � f .x/j :

The second summand is less than �=2. Since x … Bq, the first summand is alsosmaller than �=2, whenever h > 0 is small enough. So

�

1

2h

Z h

�hj f .x � t/ � f .x/jp dt

�1=p

< �;

whenever h > 0 is small enough. Thus x 2 G is a p-Lebesgue point of f , indeed. �It is easy to see that the values f .x/ and Mpf .x/ are finite if x is a p-Lebesgue

point of f 2 W.Lp; `1/.R/ .1 � p < 1/. The next theorem generalizes Lebesgue’stheorem (see Eq. (2.9.1)) and shows that the �-means converge to the function atevery Lebesgue point.

Theorem 2.9.3 Let � 2 L1.R/, 1 � p < 1 and 1=p C1=q D 1. Ifb� 2 PEq.R/, then

limT!1 ��T f .x/ D f .x/

for all p-Lebesgue points of f 2 W.Lp; `1/.R/.

Proof Now let

G.u/ WD�Z u

�uj f .x � t/ � f .x/jp dt

�1=p

.u > 0/:

Since x is a p-Lebesgue point of f , for all � > 0 there exists m 2 Z such that

Gp.u/

2u� �p if 0 < u � 2m: (2.9.2)


��T f .x/� f .x/ D Tp2�

Z

R

. f .x � t/ � f .x//b�.Tt/ dt:

This implies that

ˇ

ˇ��T f .x/� f .x/ˇ

ˇ � Tp2�

Z

R

j f .x � t/ � f .x/jˇ

ˇ

ˇ

b�.Tt/ˇ

ˇ

ˇ dt

� A1.x/C A2.x/;

where

A1.x/ D Tp2�

mCblog2 TcX

kD�1

Z

Pk.T/j f .x � t/ � f .x/j

ˇ

ˇ

ˇ

b�.Tt/ˇ

ˇ

ˇ dt

and

A2.x/ D Tp2�

1X

kDmCblog2 TcC1

Z

Pk.T/j f .x � t/ � f .x/j

ˇ

ˇ

ˇ

b�.Tt/ˇ

ˇ

ˇ dt;

where Pk.T/ was defined in (2.7.3).Similar to (2.7.4),

jA1.x/j � CT1�1=qmCblog2 TcX

kD�1

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q

�Z

Pk.T/j f .x � t/ � f .x/jp dt

�1=p

� CT1=pmCblog2 TcX

kD�1

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q

G

�

2k

T

�

:

Since 2k=T � 2mT=T � 2m, we may apply (2.9.2) to obtain

jA1.x/j � Cp�

mCblog2 TcX

kD�12k=p

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q

� Cp�

b�

PEq

:

We can see in the same way that

jA2.x/j � CT1=p1X

kDmCblog2 TcC1

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q

�Z

Pk.T/j f .x � t/ � f .x/jp dt

�1=p

:

Since Mp f .x/ and f .x/ are finite if x is a p-Lebesgue point of f , we have

Z

Pk.T/j f .x � t/ � f .x/jp dt � Cp

2k

T

Mpp f .x/C j f .x/jp

�

and so

jA2.x/j � Cp

1X

kDmCblog2 TcC12k=p

�Z

Pk

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

�1=q

Mp f .x/C j f .x/j�

:

Since blog2 Tc ! 1 as T ! 1 andb� 2 PEq.R/, we conclude that A2.x/ ! 0 asT ! 1. �

Note that W.L1; `1/.R/ contains all Lp.R/ spaces and amalgam spacesW.Lp; `q/.R/ for the full range 1 � p; q � 1. If f is continuous at a point x,then x is a p-Lebesgue point of f for every 1 � p < 1.

Corollary 2.9.4 Let � 2 L1.R/, 1 � p < 1 and 1=p C 1=q D 1. Ifb� 2 PEq.R/ andf 2 W.Lp; `1/.R/ is continuous at a point x, then

limT!1 ��T f .x/ D f .x/:

Now we are able to prove the converse of Theorem 2.9.3.

Theorem 2.9.5 Suppose that � 2 L1.R/,b� 2 L1.R/, 1 � p < 1 and 1=p C 1=q D1. If

limT!1 ��T f .x/ D f .x/ (2.9.3)

for all p-Lebesgue points of f 2 Lp.R/, thenb� 2 PEq.R/.

Proof We define the space PDp.I.0; 1// .1 � p < 1/ by taking the supremumin (2.7.5) over all 0 < r � 1 and in (2.7.6) over all k � 0. Then by (2.7.5) clearlyPDp.I.0; 1// � Lp.I.0; 1// and

�Z

I.0;1/j f jp d�

�1=p

� Ck f k PDp.I.0;1//:

The space PD0p.I.0; 1// consists of all functions f 2 PDp.I.0; 1// for which f .0/ D 0

and 0 is a p-Lebesgue point of f , in other words

limh!0

�

1

h

Z h

�hj f .u/jp du

�1=p

D 0:

We will show that PD0p.I.0; 1// is a Banach space. Let . fn/ be a Cauchy sequence

in PD0p.I.0; 1//, i.e.

k fn � fmk PD0p.I.0;1// ! 0 as n;m ! 1:

Then there exists a subsequence . f�n/ such that

f�nC1� f�n

PD0p.I.0;1// � 2�n:

Then

1X

nD0

ˇ

ˇ f�nC1� f�n

ˇ

ˇ

Lp.I.0;1//

�

1X

nD0

ˇ

ˇ f�nC1� f�n

ˇ

ˇ

PD0p.I.0;1//� 2;

thus the seriesP1

nD0ˇ

ˇ f�nC1� f�n

ˇ

ˇ is almost everywhere finite. That is to say thesequence . f�n/ is almost everywhere convergent. Let

f WD limn!1 f�n and f .0/ D 0:

For all � > 0, there exists N such that

k f � f�N k PD0p.I.0;1// �1X

nDN

f�nC1� f�n

PD0p.I.0;1// �1X

nDN

2�n < �:

If h > 0 is small enough, then

�

1

2h

Z h

�hj f�N .u/jp du

�1=p

< �:

Hence

�

1

2h

Z h

�hj f .u/jp du

�1=p

� k f � f�N k PD0p.I.0;1// C�

1

2h

Z h

�hj f�N .u/jp du

�1=p

< 2�;

whenever h is small enough. From this it follows that f 2 PD0p.I.0; 1// and 0 is a

Lebesgue point of f . Thus PD0p.I.0; 1// is a Banach space, indeed.

We get from (2.9.3) that limT!1 ��T f .0/ D 0 for all f 2 PD0p.I.0; 1//. Thus the

operators

UT W PD0p.I.0; 1// ! R; UTf WD ��T f .0/ .T > 0/

are uniformly bounded by the Banach-Steinhaus theorem. Similar to (2.7.8), wehave

supk f k

PDp.I.0;1//�1

T

ˇ

ˇ

ˇ

ˇ

Z

I.0;1/f .�t/b�.Tt/ dt

ˇ

ˇ

ˇ

ˇ

D T

b�.T�/

PEq.I.0;1//.1 � p < 1/;

where PEq.I.0; 1// is defined by taking the sumP0

kD�1 in (2.7.1). Of course, wemay suppose that f is 0 in a neighborhood of 0. Then

kUTk WD supk f k

PD0p.I.0;1//�1

T

ˇ

ˇ

ˇ

ˇ

Z

I.0;1/f .�t/b�.Tt/ dt

ˇ

ˇ

ˇ

ˇ

D Tkb�.T�/k PEq.I.0;1//� C

for all T > 0. If T D 2l then

C � T

b�.T�/

PEq.I.0;1//

D T0X

kD�12k=p

b�.T�/1Pk

q

D T0X

kD�12k=p

�Z

Pk

ˇ

ˇ

ˇ

b�.Tt/ˇ

ˇ

ˇ

qdt

�1=q

D0X

kD�12.kCl/=p

Z

PkCl

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

for all l 2 N. Hence

b�

PEq

� C;

which finishes the proof of the theorem. �Of course, the same theorem is true if (2.9.3) holds for all p-Lebesgue

points of f 2 W.Lp; `1/.R/. The next corollary follows from Theorems 2.9.3and 2.9.5.

2.10 Strong Summability 119

Corollary 2.9.6 Suppose that � 2 L1.R/,b� 2 L1.R/, 1 � p < 1 and 1=pC1=q D1. Then

limT!1 ��T f .x/ D f .x/

for all p-Lebesgue points of f 2 Lp.R/ (resp. of f 2 W.Lp; `1/.R/) if and only ifb� 2 PEq.R/.

2.10 Strong Summability

As we mentioned earlier, the Fejér means of the trigonometric Fourier series of anintegrable function converge almost everywhere to the function, i.e.

1

n C 1

nX

kD0

sk f .x/ � f .x/�

! 0 as n ! 1

for almost every x 2 T. The set of convergence is characterized as the Lebesguepoints of f .

Hardy and Littlewood [170] considered the so-called strong summability andverified that the strong means

1

n C 1

nX

kD0jsk f .x/ � f .x/jq

tend to 0 at each Lebesgue point of f as n ! 1, whenever f 2 Lp.T/ and1 < p < 1, 0 < q < 1 (for Fourier transforms see Giang and Móricz[139]). This result does not hold for p D 1 (see Hardy and Littlewood [171]).However, the strong means tend to 0 almost everywhere for all f 2 L1.T/. Thisis due to Marcinkiewicz [242] for q D 2 and to Zygmund [399] for all q > 0

(see also the book of Bary [16]). Later Gabisoniya [121, 122] (see also Rodin[278]) characterized the set of convergence as the so-called Gabisoniya points.These results are unknown for Fourier transforms. In this section we present theanalogues of these results for Fourier transforms and for functions from the Wieneramalgam spaces. Some of the results are given here without proofs. The proofscan be found later in Sect. 5.5 since to the proofs we need some multi-dimensionalresults formulated in Chap. 5.

In this section we use other conditions about � . Namely, suppose that � is evenand absolutely continuous. Suppose further that

�.0/ D 1;

Z 1

0

.t _ 1/dj� 0.t/j dt < 1; limt!1 td�.t/ D 0 (2.10.1)

for some d 2 N, where _ denotes the maximum and ^ the minimum. Denote by

soc t WD�

cos t; if d is even;sin t; if d is odd

(2.10.2)

and assume that for all u > 0 and for some 0 < ˛ < 1,

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/ ti .soc /.i/.tu/ dt

ˇ

ˇ

ˇ

ˇ

� Cu�˛ .i D 0; : : : ; d � 1/: (2.10.3)

We say that f is locally bounded at x if there exists a neighborhood of x such thatf is bounded on this neighborhood. The first version of our strong summabilityresults reads as follows. Note that the Dirichlet integral st f is well defined whenf 2 W.L1; `q/.R/ for some 1 � q < 1. Theorems 2.10.1 and 2.10.3 will be provedlater in Sect. 5.5.

Theorem 2.10.1 Suppose that (2.10.1) and (2.10.3) hold for some d 2 N and 0 <˛ < 1. Let f 2 W.L1; `q/.R/ for some 1 � q < 1. If xj is a Lebesgue point of fand f is locally bounded at xj for all j D 1; : : : ; d, then

limT!1

�1T

Z 1

0

� 0 t

T

�dY

jD1.st f .xj/� f .xj// dt D 0:

Writing x1 D : : : D xd D x, we obtain

Corollary 2.10.2 Suppose that (2.10.1) and (2.10.3) hold for some even d 2 N and0 < ˛ < 1. Let f 2 W.L1; `q/.R/ for some 1 � q < 1. If x 2 R is a Lebesguepoint of f and f is locally bounded at x, then

limT!1

�1T

Z 1

0

� 0 t

T

�

jst f .x/ � f .x/jd dt D 0:

If f is almost everywhere locally bounded, then the corollary holds almosteverywhere. It is not true that an integrable function is almost everywhere locallybounded. Let us denote the Cantor set of Lebesgue measure 1=2 by H � Œ0; 1�. Weobtain H in the following way. In the first step we omit the interval I11 of measure1=4 from the middle of Œ0; 1�. In the second step we omit the intervals I12 and I22 oflength 1=16 from the middle of the remaining two intervals, in the kth step we omitthe intervals I1k ; : : : ; I

2k�1

k of length 1=4k. We define the function f by f WD 0 on H

and f .x/ WD .x � ajk/

�1=2=k2 if x 2 Ijk D .aj

k; bjk/. Then f is integrable and

Z 1

0

f d� D 2

1X

kD1

2k�1X

jD1

.bjk � aj

k/1=2

k2D 2

1X

kD12k�1 1

k22kD �2

6:

On the other hand f is not almost everywhere locally bounded, because for everyx 2 H and every neighborhood of x there are a j

k’s contained in this neighborhood,and so f is not locally bounded at x.

We will extend Corollary 2.10.2 to each f 2 W.L1; `q/.R/ .1 � q < 1/ later.For the convergence of f 2 W.Lp; `q/.R/ .1 < p < 1; 1 � q < 1/ at p-Lebesguepoints we get the following result. Note that W.Lp; `p/.R/ D Lp.R/.

Theorem 2.10.3 Let 1 < p < 1, 1 � q < 1 and f 2 W.Lp; `q/.R/. Supposethat (2.10.1) and (2.10.3) hold for some d 2 N and 0 < ˛ < 1. If xj is a p-Lebesgue point of f for all j D 1; : : : ; d, then

limT!1

�1T

Z 1

0

� 0 t

T

�dY


Corollary 2.10.4 Let 1 < p < 1, 1 � q < 1 and f 2 W.Lp; `q/.R/. Supposethat (2.10.1) and (2.10.3) hold for some even d 2 N and 0 < ˛ < 1. If x 2 R is ap-Lebesgue point of f , then

limT!1

�1T

Z 1

0

� 0 t

T

�


Obviously, the convergence holds almost everywhere. We will extend this resultto p D 1 as follows. To this end we introduce the Gabisoniya points.

Definition 2.10.5 A point x 2 R is said to be a Gabisoniya point of f 2W.L1; `1/.R/ if for all 1 < < 1,

limT!1

1X

iD1

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

D 0:

It is easy to see that every Gabisoniya point is also a Lebesgue point. In thissection we are able to prove that almost every point is a Gabisoniya point.

Theorem 2.10.6 Almost every point x 2 R is a Gabisoniya point of f 2W.L1; `1/.R/.

Before proving this theorem, we introduce some definitions and lemmas. A setP is said to be a perfect set if it is closed and has no isolated points. If P � .0; 1/

is a perfect set, then .0; 1/ n P is open and so it can be decomposed into the unionof countably many disjoint open intervals. These intervals are called the adjacent

intervals of P with respect to the interval .0; 1/. Recall that for an interval I, cIdenotes the interval with the same centre and with length c jIj .c > 0/.Lemma 2.10.7 Let P � .0; 1/ be a perfect set with adjacent intervals In D.˛n; ˇn/ .n 2 N/ and let

f .t/ WD�

ˇn � ˛n; t 2 1=3InI0; else.

Then for all 1 < < 1,

Z 1

0

f .t/�1

jt � xj dt < 1 for a.e. x 2 P:


Z 1

0

Z 1

0

1P.x/f .t/�1

jt � xj dt dx D1X

nD0

Z

1=3In

f .t/�1Z 1

0

1P.x/

jt � xj dx dt:

If x 2 P and t 2 1=3In, then jt � xj � 1=3jInj and it follows that

Z 1

0

1P.x/

jt � xj dx D 2

Z 1

1=3jInj1

udu � C

jInj�1 :

Hence

Z 1

0

Z 1

0

1P.x/f .t/�1

jt � xj dt dx �1X

nD0j1=3Inj < 1

and the lemma follows from Fubini’s theorem. �Recall that %.x; In/ denotes the distance between x and In.

Lemma 2.10.8 Let P � .0; 1/ be a perfect set with adjacent intervals In D.˛n; ˇn/ .n 2 N/. Then for all 1 < < 1,

1X

nD0

.ˇn � ˛n/

%.x; In/< 1 for a.e. x 2 P:

Proof Suppose that x 2 P n [rIn for some fixed r > 1. For a given interval In

suppose that t 2 1=3In and x > ˇn. Then x � t � x � ˛n D x � ˇn C jInj. IfC � 1 � 2=.r � 1/, then

x � ˇn C jInj � C.x � ˇn/ D C%.x; In/:

Indeed, since x … rIn,

x � ˇn � r � 1

2jInj;

which shows the preceding inequality. By Lemma 2.10.7,

1 >

Z 1

0

f .t/�1

jt � xj dt D1X

nD0

Z

1=3In

f .t/�1

jt � xj dt � 1

3C

1X

nD0

.ˇn � ˛n/

%.x; In/:

Thus the lemma holds for almost every x 2 P n [rIn. The lemma follows from thefact that r > 1 is arbitrary. �

The next lemma is due to Gabisoniya [121] for f 2 L1.T/.

Lemma 2.10.9 Suppose that f 2 W.L1; `1/.R/ and 1 < < 1. For almost everypoint x 2 R, we have

limT!1

bTcX

iD1

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

D 0:

Proof We may suppose that x 2 .0; 1/. Since almost every point is a Lebesgue pointof f 2 W.L1; `1/.R/, we have

limn!1 n

Z 1=n

�1=nj f .x � u/� f .x/j du D 0

for almost every point x. By Egorov’s theorem, for all positive integer k, we can finda set Uk � .�1; 2/ such that jUkj > 3 � 1=k and the previous convergence holdsuniformly on Uk. For T > 1 there exists n 2 N such that 1=2n � 1=T < 1=n and so

TZ 1=T

�1=Tj f .x � u/� f .x/j du � 2n

Z 1=n

�1=nj f .x � u/� f .x/j du:

Thus

limT!1 T

Z 1=T

�1=Tj f .x � u/� f .x/j du D 0

uniformly on Uk, in other words, for arbitrary � > 0 there exists T0 > 0 such thatfor all T > T0

supx2Uk0

TZ 1=T

�1=Tj f .x � u/� f .x/j du < �: (2.10.4)

It is easy to see that there exists a perfect set Pk � Uk such that

jPkj > jUkj � 1=k and supx2Pk

j f .x/j DW Bk f < 1:

Let Ek � Pk denote the set of those points x 2 Pk for which

1X

nD0

ˇ.k/n � ˛

.k/n

�

�

x; I.k/n

� < 1 (2.10.5)

for all 1 < < 1, where I.k/n D

˛.k/n ; ˇ

.k/n

�

are the adjacent intervals of Pk �.�1; 2/ with respect to the interval .�1; 2/. By Lemma 2.10.8, jEkj D jPkj. Then

ˇ

ˇ

ˇ

ˇ

ˇ

1[

kD0Ek

\

.0; 1/

ˇ

ˇ

ˇ

ˇ

ˇ

D 1:

For x 2 [1kD0Ek \ .0; 1/ let us fix a number k0 for which x 2 Ek0 � Pk0 . Since the

series in (2.10.5) is convergent, for arbitrary positive � there exists N1 such that

1X

rDN1

.ˇr � ˛r/

%.x; Ir/< �; (2.10.6)

where Ir D .˛r; ˇr/ are the adjacent intervals of Pk0 � .�1; 2/ with respect to theinterval .�1; 2/. Two cases are possible for each 1 � i � bTc: Ji \ Pk0 ¤ ; orJi \ Pk0 D ;, where Ji D Œx � i=T; x � .i � 1/=T�.

Case 1. If Ji \ Pk0 ¤ ;, then there is a number 0 < � < 1=T such that

x � i=T C � 2 Pk0 :

Then by (2.10.4),

TZ i=T

.i�1/=Tj f .x � u/� f .x/j du

D TZ �

��1=Tj f .x � i=T C � � t/ � f .x/j du

� TZ 1=T

�1=Tj f .x � i=T C � � t/ � f .x � i=T C �/j du

C TZ 1=T

�1=Tj f .x � i=T C �/ � f .x/j du

� � C 4Bk0 f (2.10.7)

for T > T0. Let us denote byP.1/

i the sum over all indices 1 � i � bTc for which

Ji \ Pk0 ¤ ; andP.2/

i for which Ji \ Pk0 D ;. We can choose an N 2 N such that

.� C 4Bk0 f /

. � 1/N�1 < �:

Then by (2.10.7),

.1/X

i

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

D0

@

.1/X

i<N

C.1/X

i�N

1

A

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

�

TZ N=T

0

j f .x � u/� f .x/j du

!

C .� C 4Bk0 f /X

i�N

1

i

� � C .� C 4Bk0 f /

. � 1/N�1 < 2�

if T is large enough.Case 2. If Ji \Pk0 D ;, then there exists r 2 N such that Ji � Ir D .˛r; ˇr/. Thus

˛r < x � i=T < x � .i � 1/=T < ˇr < x:

Therefore T=i � 1=.x � ˇr/. Let N2 be a natural number for which

1X

iDN2C1

1

i<

�

4Bk0 f :

Of courseP1

rD0.ˇr �˛r/ < 1 and so there exists N3 � N1 such that for all r > N3,

ˇr � ˛r < 1=T0:

Since ˇr 2 Pk0 , by (2.10.4),

1

ˇr � ˛r

Z ˇr�˛r

0

j f .ˇr � u/ � f .ˇr/j du < � (2.10.8)

for r > N3. Then

.2/X

i

T

i

Z i=T

.i�1/=Tj f .x � u/ � f .x/j du

!

D0

@

.2/X

i�N2

C.2/X

i>N2

1

A

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

�

TZ N2=T

0

j f .x � u/� f .x/j du

!

C1X

rD0

X

i>N2;Ji�Ir

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

� � C 21X

rD0

X

i>N2;Ji�Ir

T

i

Z i=T

.i�1/=Tj f .x � u/� f .ˇr/j du

!

(2.10.9)

C 21X

rD0

X

i>N2;Ji�Ir

T

i

Z i=T

.i�1/=Tj f .ˇr/� f .x/j du

!

if T is large enough. Since x; ˇr 2 Pk0 , the last term can be estimated as follows:

21X

rD0

X

i>N2;Ji�Ir

T

i

Z i=T

.i�1/=Tj f .ˇr/ � f .x/j du

!

� 4Bk0 f X

i>N2

1

i< �:

For the second term of (2.10.9), we obtain

21X

rD0

X

i>N2;Ji�Ir

T

i

Z i=T


!

D 2

0

@

N3X

rD0C

1X

rDN3C1

1

A

X

i>N2;Ji�Ir

T

i

Z i=T


!

� 2 max0�r�N3

1

.x � ˇr/sup

i>N2;Ji�Ir

Z x�.i�1/=T

x�i=Tj f .t/ � f .ˇr/j dt

!�1

N3X

rD0

X

i>N2;Ji�Ir

Z x�.i�1/=T


C 21X

rDN3C1

1

.x � ˇr/

0

@

X

i>N2;Ji�Ir

Z x�.i�1/=T


1

A

:

For an integrable function g, the measureR

� g d� is uniformly continuous withrespect to �. From this it follows that

Z x�i=T

x�.i�1/=Tj f .t/ � f .ˇr/j dt < �=C

for all i if T is large enough. Since

N3X

rD0

X

i>N2;Ji�Ir

Z x�i=T

x�.i�1/=Tj f .t/ � f .ˇr/j dt �

N3X

rD0

Z ˇr

˛r

j f .t/ � f .ˇr/j dt < 1;

using (2.10.6) and (2.10.8), we conclude

21X

rD0

X

i>N2;Ji�Ir

T

i

Z i=T


!

� � C 21X

rDN3C1

1

.x � ˇr/

Z ˇr

˛r

j f .t/ � f .ˇr/j dt

!

� � C 21X

rDN3C1

1

.x � ˇr/

Z ˇr�˛r

0

j f .ˇr � u/� f .ˇr/j du

!

� � C 2�1X

rDN3C1

.ˇr � ˛r/

.x � ˇr/< 2�

if 0 < � < 1=2 and T is large enough. �Now we can prove Theorem 2.10.6.

Proof of Theorem 2.10.6 We can prove similarly as in Lemma 2.10.9 that

limT!1

kbTcX

iD1

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

D 0

for almost every x 2 R and for every fixed k 2 N. Observe that

1X

iDkbTcC1

T

i

Z i=T

.i�1/=Tj f .x/j du

!

�1X

iDkbTcC1

1

ij f .x/j � C.kT/1� j f .x/j :


On the other hand, by Abel rearrangement,

.kC1/bTcX

iDkbTcC1

T

i

Z i=T

.i�1/=Tj f .x � u/j du

!

D.kC1/bTc�1X

iDkbTcC1

�

T

i� T

.i C 1/

�

Si C T

..k C 1/bTc/ S.kC1/bTc

� C.kC1/bTc�1X

iDkbTcC1T i��1Si C C

.k C 1/S.kC1/bTc;

where

Si WDiX

jDkbTcC1

Z j=T

. j�1/=Tj f .x � u/j du

!

�0

@

iX

jDkbTcC1

Z j=T

. j�1/=Tj f .x � u/j du

1

A

� C k f kW.L1;`1/

for kbTc C 1 � i � .k C 1/bTc. Hence

.kC1/bTcX

iDkbTcC1

T

i

Z i=T

.i�1/=Tj f .x � u/j du

!

� Ck� k f kW.L1;`1/ :

Taking into account these estimations, we conclude

1X

iD1

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

�k0bTcX

iD1

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

C1X

kDk0

.kC1/bTcX

iDkbTcC1

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

�k0bTcX

iD1

T

i

Z i=T

.i�1/=Tj f .x � u/� f .x/j du

!

C Ck1�0 k f kW.L1;`1/ C C.k0T/1� j f .x/j ;

which is small enough if k0 and T are large enough. �

The next theorem will also be proved in Sect. 5.5.

Theorem 2.10.10 Suppose that (2.10.1) and (2.10.3) hold for some d 2 N and1 < ˛ < 1. If f 2 W.L1; `q/.R/ for some 1 � q < 1 and xj is a Gabisoniya pointof f for all j D 1; : : : ; d, then

limT!1

�1T

Z 1

0

� 0 t

T

�dY


Corollary 2.10.11 Suppose that (2.10.1) and (2.10.3) hold for some even d 2 N

and 1 < ˛ < 1. If f 2 W.L1; `q/.R/ for some 1 � q < 1 and x 2 R is aGabisoniya point of f , then

limT!1

�1T

Z 1

0

� 0 t

T

�


Since almost every point is a Gabisoniya point of f 2 W.L1; `q/.R/ .1 � q <1/, the convergence holds almost everywhere. Remark that W.L1; `p/.R/ Lp.R/

for all 1 � p < 1. Taking the Fejér summability, we obtain the followingcorollaries from the results above.

Corollary 2.10.12 Suppose that f 2 W.L1; `q/.R/ for some 1 � q < 1 and d iseven. If x is a Lebesgue point of f and f is locally bounded at x, then

limT!1

1

T

Z T

0

jst f .x/� f .x/jd dt D 0:

Corollary 2.10.13 Suppose that f 2 W.Lp; `q/.R/ for some 1 < p < 1, 1 � q <1 and d is even. If x is a p-Lebesgue point of f , then

limT!1

1

T

Z T

0


Though the Fejér summation does not satisfy the condition of Corollary 2.10.11,because ˛ D 1 in this case, the next corollary is valid.

Corollary 2.10.14 If d is even, f 2 W.L1; `q/.R/ for some 1 � q < 1 and x is aGabisoniya point of f , then

limT!1

1

T

Z T

0


Proof It is easy to see that �.t/ WD e�t satisfies the condition of Corollary 2.10.11with ˛ D 2 if d is even (see also Example 2.11.8). Then the proof follows from theinequality 1=e � e�t=T on the interval Œ0;T�. �

Note that for an odd d and for �.t/ WD e�t, we have ˛ D 1. Of course,the corollary holds almost everywhere. Marcinkiewicz [242] and Zygmund [399]proved that the convergence holds almost everywhere for all f 2 L1.T/, but it doesnot hold at each Lebesgue point of f (see Hardy and Littlewood [171]). However, iff is almost everywhere locally bounded, resp. if f 2 Lp.R/ or W.Lp; `q/.R/ .1 0, � is non-increasing, (2.10.1) and (2.10.3)are satisfied for all d. Under the same conditions as in Corollaries 2.10.4 or 2.10.11,respectively, we get that

limT!1

�1T

Z 1

0

� 0 t

T

�

jst f .x/� f .x/jr dt D 0:

Proof For a fixed r choose an even d > r. Since � 0 � 0, by Hölder’s inequality

1

T

Z 1

0

ˇ

ˇ

ˇ�0 t

T

�ˇ

ˇ

ˇ jst f .x/ � f .x/jr dt

D 1

T

Z 1

0

ˇ

ˇ

ˇ�0 t

T

�ˇ

ˇ

ˇ

r=d

jst f .x/ � f .x/jrˇ

ˇ

ˇ�0 t

T

�ˇ

ˇ

ˇ

1�r=d

dt

� 1

T

�Z 1

0

ˇ

ˇ

ˇ� 0 t

T

�ˇ

ˇ

ˇ jst f .x/ � f .x/jd dt

�r=d �Z 1

0

ˇ

ˇ

ˇ� 0 t

T

�ˇ

ˇ

ˇ dt

�1�r=d

� C

�

1

T

Z 1

0

ˇ

ˇ

ˇ� 0 t

T

�ˇ

ˇ

ˇ jst f .x/ � f .x/jd dt

�r=d

;

which tends to zero by Corollary 2.10.4 or 2.10.11. �Similarly, for the strong Fejér summation we have

Corollary 2.10.16 Suppose that r > 0. Under the same conditions as in Corol-lary 2.10.13 or 2.10.14, respectively, we get that

limT!1

1

T

Z T

0

jst f .x/ � f .x/jr dt D 0:

Proof Here we chose an even d with d > r. Then the result follows fromCorollary 2.10.13 or 2.10.14. �

2.11 Some Summability Methods

In this section we consider some summability methods as special cases of the �-summation. Of course, there are a lot of other summability methods which could beconsidered as special cases.

2.11 Some Summability Methods 131

Lemma 2.11.1 Let � 2 L1.R/ \ C0.R/ be even and �.k/ absolutely continuousfor k D 0; 1 : : : ; i, where i � 1. Suppose that limt!C0.ti�.t//.k/ D 0 andlimt!1.ti�.t//.k/ D 0 for 0 � k < i and, moreover, limt!C0.ti�.t//.i/ 2 R andlimt!1.ti�.t//.i/ D 0. If .ti�.t//.iC1/ is integrable, thenb� 2 L1.R/ and

ˇ

ˇ

ˇ

b�.i/.x/ˇ

ˇ

ˇ � C

xiC1 .x 2 .0;1//:

Proof If i is even, then we get by integrating by parts that

ˇ

ˇ

ˇ

b�.i/.x/ˇ

ˇ

ˇ D 2p2�

ˇ

ˇ

ˇ

ˇ

Z 1

0

ti�.t/ cos tx dt

ˇ

ˇ

ˇ

ˇ

D C

x

ˇ

ˇ

ˇ

ˇ

Z 1

0

.ti�.t//0 sin tx dt

ˇ

ˇ

ˇ

ˇ

D : : :

D C

xiC1ˇ

ˇŒ.ti�.t//.i/ cos tx�10ˇ

ˇC C

xiC1

ˇ

ˇ

ˇ

ˇ

Z 1

0

.ti�.t//.iC1/ cos tx dt

ˇ

ˇ

ˇ

ˇ

:

Of course, the lemma can be proved in the same way if i is odd. �

Lemma 2.11.2 Let � 2 L1.R/\C0.R/ be even and � , � 0 be absolutely continuous.If limt!C0 � 0 2 R, limt!1 � 0 D 0 and if � 00 is integrable, thenb� 2 L1.R/ and

ˇ

ˇ

ˇ

b�.x/ˇ

ˇ

ˇ � C

x2.x 2 .0;1//:

Proof Similarly as above,

ˇ

ˇ

ˇ

b�.x/ˇ

ˇ

ˇ D C

x

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/ sin tx dt

ˇ

ˇ

ˇ

ˇ

� C

x2ˇ

ˇŒ� 0.t/ cos tx�10ˇ

ˇC C

x2

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 00.t/ cos tx dt

ˇ

ˇ

ˇ

ˇ

;

which proves also the integrability ofb� . �Using these lemmas we can show that in all the following examples � 2

L1.R/ \ C0.R/,b� 2 L1.R/ and b� 2 PE1.R/. Moreover, (2.10.1) holds as well. InExamples 2.11.3–2.11.10 condition (2.8.1) is true for N D 0 and ˇ D 2. The same isvalid in the last example if 1 � ı; < 1. Examples 2.11.3–2.11.7 satisfy (2.10.3)with ˛ D 1. Examples 2.11.8–2.11.10 satisfy (2.10.3) with ˛ D 2 for an even dand with ˛ D 1 for an odd d. Moreover, if 1 � ı; < 1 in Example 2.11.11, then˛ D 1. The elementary computations are left to the reader.

Example 2.11.3 (Fejér Summation) Let

�.t/ WD�

1 � jtj; if jtj � 1I0; if jtj > 1:

Example 2.11.4 (de La Vallée-Poussin Summation) Let

�.t/ D8

<

:

1; if jtj � 1=2I�2jtj C 2; if 1=2 < jtj � 1I0; if jtj > 1:

Example 2.11.5 (Jackson-de La Vallée-Poussin Summation) Let

�.t/

8

<

:

1 � 3t2=2C 3jtj3=4; if jtj � 1I.2 � jtj/3=4; if 1 < jtj � 2I0; if jtj > 2:

The next example generalizes Examples 2.11.3–2.11.5.

Example 2.11.6 Let 0 D ˛0 < ˛1 < : : : < ˛m and ˇ0; : : : ; ˇm .m 2 N/ be realnumbers, ˇ0 D 1, ˇm D 0. Suppose that � is even, �.˛j/ D ˇj . j D 0; 1; : : : ;m/,�.t/ D 0 for t � ˛m, � is a polynomial on the interval Œ˛j�1; ˛j� . j D 1; : : : ;m/.

Example 2.11.7 (Rogosinski Summation) Let

�.t/ D�

cos�t=2; if jtj � 1C 2jI0; if jtj > 1C 2j:

Example 2.11.8 (Weierstrass Summation) Let

�.t/ D e�jtj

for some 1 � < 1. Note that if D 1, then we obtain the Abel summation.

Example 2.11.9 Let

�.t/ D e�.1Cjtjq/

with 1 � q < 1; 0 < < 1.

Example 2.11.10 (Picard and Bessel Summations) Let

�.t/ D .1C jtj /�ı

with 0 < ı < 1; 1 � < 1; ı > d.

2.11 Some Summability Methods 133

Example 2.11.11 (Riesz Summation) Let

�.t/ WD�

.1 � jtj /ı; if jtj � 1I0; if jtj > 1:

for some 0 < ı < 1; 1 � < 1.If in the last example 0 < ı � 1 � < 1, then (2.8.1) holds with N D 0,

ˇ D ı C 1 and (2.10.3) with ˛ D ı (for an elementary proof see [355, p. 205, p.259]). Later in Sect. 5.3.3 we will show that (2.8.1) holds as well for 1 < ı < 1,then N < ı � N C 1 and ˇ D ı C 1. Moreover, Examples 2.11.8 and 2.11.10satisfy (2.8.7) for all N 2 N.

Part IIMulti-Dimensional Hardy Spaces

and Fourier Transforms

Chapter 3Multi-Dimensional Hardy Spaces

In this chapter, two types of the multi-dimensional classical Hardy spaces, namelythe H�

p .Rd/ and Hp.R

d/ spaces, are introduced. All the results of Chap. 1, amongstothers, inequalities, atomic decompositions, interpolation theorems, boundednessresults are proved for these spaces. Basically, the results for H�

p .Rd/ are very similar

to those for the one-dimensional Hp.R/ spaces studied in Chap. 1, so we omit thecorresponding proofs. However, the proofs for Hp.R

d/ are different from the one-dimensional version requiring new ideas. We also study some generalizations of theHardy-Littlewood maximal function for multi-dimensional functions.

3.1 Multi-Dimensional Maximal Functions

We introduce some versions of the Hardy-Littlewood maximal function and thestrong maximal function. As in the one-dimensional case, the Hardy-Littlewoodmaximal function is bounded on Lp.R

d/ for 1 < p � 1 and it is of weak type .1; 1/.However, the strong maximal function is also bounded on Lp.R

d/ for 1 < p � 1,but it is not of weak type .1; 1/.

3.1.1 Hardy-Littlewood Maximal Functions

Let us fix d � 2, d 2 N. For a set Y ¤ ; let Yd be its Cartesian product Y : : :Y

taken with itself d-times. The Lp.Rd/ spaces are defined as in the one-dimensional

case. Denote logC u WD max.0; log u/.


137

138 3 Multi-Dimensional Hardy Spaces

Definition 3.1.1 For k 2 N, a measurable function f is in the set Lp.log L/k.Rd/

.1 � p < 1/ if

k f kLp.log L/k WD�Z

Rdj f jp.logC j f j/k d�

�1=p

< 1:

If p D 1, then set L1.log L/k.Rd/ D L1.Rd/.For k D 0 we get back the Lp.R

d/ spaces. For X � R the set Lp.log L/k.Xd/ isdefined analogously. If X has finite measure, then for all k 2 P and p > 1,

L1.Xd/ L1.log L/k�1.Xd/ L1.log L/k.Xd/ Lp.X

d/:

Definition 3.1.2 A measurable function f belongs to the Wiener amalgam spaceW.Lp; `q/.R

d/ .1 � p; q � 1/ if

k f kW.Lp ;`q/WD0

@

X

k2Zd

k f .� C k/kqLpŒ0;1/d

1

A

1=q

< 1;

with the obvious modification for q D 1. Replacing here the space LpŒ0; 1/d by

Lp;1Œ0; 1/d, or by Lp.log L/kŒ0; 1/d, we get the definition of W.Lp;1; `q/.Rd/ and

W.Lp.log L/k; `q/.Rd/. The spaces W.C; `q/.R

d/ and W.Lp; c0/.Rd/ are definedanalogously .1 � p; q � 1/.

The same embeddings are true for the Wiener amalgam spaces as in the one-dimensional case. For x D .x1; : : : ; xd/ 2 R

d and u D .u1; : : : ; ud/ 2 Rd set

u � x WDdX

kD1ukxk; kxkp WD

dX

kD1jxkjp

!1=p

.1 � p < 1/

and

kxk1 WD supkD1;:::;d

jxkj ; jxj WD kxk2 :

The balls with centre c and radius h are denoted by B.c; h/ .c 2 Rd; h > 0/, i.e.

B.c; h/ WD fx 2 Rd W jx � cj < hg:

As in the one-dimensional case the Hardy-Littlewood maximal function can begiven by

Mpf .x/ D supx2I

�

1

jIjZ

Ij f jp d�

�1=p

.x 2 Rd/;

3.1 Multi-Dimensional Maximal Functions 139

where the supremum is taken over all cubes with sides parallel to the axes. However,in this section we will rather use the next equivalent centred version.

Definition 3.1.3 For a locally integrable function f 2 Llocp .R

d/ the Hardy-Littlewood maximal function is defined by

Mpf .x/ WD suph>0

�

1

.2h/d

Z h

�h� � �Z h

�hj f .x � s/jp ds

�1=p

:

Both definitions can be given with the help of balls instead of cubes. The nexttheorem can be proved exactly as in the one-dimensional case Theorem 1.2.4 andCorollaries 1.2.5 and 2.7.9.

Theorem 3.1.4 If 1 � p < 1, then

sup�>0

��.Mpf > �/1=p � Ck f kp . f 2 Lp.Rd//;

Mpf

r� Cr k f kr . f 2 Lr.R

d/; p < r � 1/

and

supk2Zd

sup�>0

��.Mpf > �; Œk; k C 1//1=p D

Mpf

W.Lp;1;`1/� Cp k f kW.Lp;`1/

for all f 2 W.Lp; `1/.Rd/. Moreover, for every p < r � 1,

Mpf

W.Lr ;`1/� Cr k f kW.Lr ;`1/ . f 2 W.Lr; `1/.Rd//:

Lebesgue’s differentiation theorem can be formulated as follows.

Corollary 3.1.5 If f 2 Lloc1 .R

d/, then

limh!0

1

.2h/d

Z h

�h� � �Z h

�hf .x � s/ ds D f .x/ a.e. x 2 R:

Note that Lloc1 .R

d/ contains the spaces Lp.Rd/ and W.Lp; `q/.R

d/ for all 1 �p; q � 1.

We are going to generalize the Hardy-Littlewood maximal function around thediagonals. In order to simplify the discussion, we fix d D 2 in the first step. Let .h/ and �.h/ be two continuous functions of h � 0, strictly increasing to 1 and 0at h D 0. Let

M.1/; ;�p f .x; y/ WD sup

h>0

1

4 .h/�.h/

Z .h/

� .h/

Z �.h/

��.h/j f .x � s; y � t/jp ds dt

!1=p

;

where f 2 Llocp .R

2/. If .h/ D �.h/ D h, then we get back the usual Hardy-Littlewood maximal function. For p D 1, we write simply Mf and M.1/; ;� f . We canprove in the same way as in Theorem 3.1.4 that M.1/; ;� has the same properties, i.e.for 1 � p < 1,

sup�>0

��.M.1/; ;�p f > �/1=p � C k f kp . f 2 Lp.R

2// (3.1.1)

and

M.1/; ;�p f

r� Crk f kr . f 2 Lr.R

2/; p < r � 1/; (3.1.2)

where the constants C and Cp are independent of and �.For some � > 0 and f 2 Lloc

p .R2/ let

M.1/p f .x; y/

WD supi;j2N;h>0

2��.iCj/

1

4 � 2iCjh2

Z 2ih

�2ih

Z 2jh

�2jhj f .x � s; y � t/jp ds dt

!1=p

:

We modify slightly the definition of this maximal function. Let

M.2/; ;�p f .x; y/ WD sup

h>0

1

4 .h/�.h/

Z .h/

� .h/

Z sC�.h/

s��.h/j f .x � s; y � t/jp dt ds

!1=p

and

M.2/p f .x; y/

WD supi;j2N;h>0

2��.iCj/

1

4 � 2iCjh2

Z 2ih

�2ih

Z sC2jh

s�2jhj f .x � s; y � t/jp dt ds

!1=p

:

Moreover, set

M.3/p f .x; y/

WD supi;j2N;h>0

2��.iCj/

1

4 � 2iCjh2

Z 2ih

�2ih

Z �sC2jh

�s�2jhj f .x � s; y � t/jp dt ds

!1=p

:

Note that in M.1/p f we take the supremum over rectangles with sides parallel to the

axes and in M.2/p f and M.3/

p f over parallelograms with sides parallel to one of theaxes and to one of the diagonals of the square Œ0; 1�2. With the same proof we cansee that (3.1.1) and (3.1.2) hold also for M.2/; ;�

p f .

Definition 3.1.6 For d D 2 and � > 0, we define the modified maximal function

M�p f .x; y/ WD M.1/

p f .x; y/C M.2/p f .x; y/C M.3/

p f .x; y/:

For simplicity, we omit the notation � and write Mpf . We use the notations

M.i/f WD M.i/1 f .i D 1; 2; 3/ and Mf WD M1 f .

Theorem 3.1.7 For 1 � p < 1 and � > 0,

sup�>0

��.Mpf > �/1=p � Ck f kp . f 2 Lp.R2//;

Mpf

r� Crk f kr . f 2 Lr.R

2/; p < r � 1/

and

Mpf

W.Lp;1;`1/� Ck f kW.Lp ;`1/ . f 2 W.Lp; `1/.R2//;

Mpf

W.Lr ;`1/� Crk f kW.Lr ;`1/ . f 2 W.Lr; `1/.R2/; p < r � 1/:

Proof Applying inequality (3.1.1) to .h/ D 2ih and �.h/ D 2jh, we obtain

��.M.1/f > �/ � �

1X

iD0

1X

jD0�.M.1/; ;� f > 2�.iCj/�/

� C1X

iD0

1X

jD02��.iCj/k f k1

� Ck f k1for all f 2 L1.R2/ and � > 0. The inequality

M.1/f

p� Cpk f kp . f 2 Lp.R

2/; 1 < p � 1/

can be shown similarly. We can see in the same way that these inequalities alsohold for M.i/f , i D 2; 3. The proof can be finished in the usual way as inCorollary 1.2.8. �

Let us turn to the higher dimensional case, i.e. let d � 3. Under a diagonalwe understand a diagonal of the unit cube Œ0; 1�d. Let us denote by P2i1h;:::;2id h aparallelepiped with side lengths 2i1C1h; : : : ; 2idC1h, .h > 0; i D .i1; : : : ; id/ 2 N

d/,whose centre is the origin and whose sides are parallel to the axes and/or to thediagonals.


Definition 3.1.8 For some � > 0 and f 2 Llocp .R

d/, the modified maximal functionis given by

M�p f .x/

WD supP2i1 h;:::;2id h

;i2Nd ;h>0

2��kik1

1ˇ

ˇP2i1h;:::;2id h

ˇ

ˇ

Z

P2i1 h;:::;2id h

j f .x � s/jp ds

!1=p

;

where the supremum is taken over all parallelepipeds P2i1h;:::;2id h .i 2 Nd; h > 0/

just defined. We omit the notation � and write simply Mpf .If we take the supremum only over all cubes with sides parallel to the axes and

� D 0, we get back the definition of the Hardy-Littlewood maximal function Mpf .Obviously, M�1 f � M�2 f for �1 > �2 > 0. It is easy to see that

Mpf .x/ � supi2Nd ;h>0

2��kik1

1

.2h/d2kik1

Z 2i1h

�2i1h

Z ı1s1C2i2h

ı1s1�2i2h� � �Z ıd�1.s1�s2��sd�1/C2id h

ıd�1.s1�s2��sd�1/�2id hj f .x � s/jp ds

!1=p

;

where ıi 2 f0; 1g .i D 1; : : : ; d/. Taking the supremum in the definition of Mpfover all parallelepipeds whose sides are parallel to the axes, we obtain the definitionof M.1/

p f :

M.1/p f .x/ WD sup

i2Nd ;h>0

2��kik1

1

.2h/d2kik1

Z 2i1h

�2i1h� � �Z 2id h

�2id hj f .x � s/jp ds

!1=p

:

For p D 1, we write simply M.1/f and Mf , respectively. Then Theorem 3.1.7 canbe shown for higher dimensions in the same way.

3.1.2 Strong Maximal Functions

For the strong maximal function we need new Wiener amalgam spaces. Let.i1; : : : ; id/ be a permutation of .1; : : : ; d/.

Definition 3.1.9 A measurable function f belongs to the iterated Wiener amalgamspace WI.Lp; `1/.Rd/ .1 � p � 1/ if

k f kWI .Lp;`1/ WD sup.i1;:::;id/

supni12Z

Z ni1C1

ni1

� � � supnid 2Z

Z nid C1

nid

j f .x/jp dxid � � � dxi1

!1=p

is finite. If we replace j f .x/jp by j f .x/jp.logC j f .x/j/k in the previous integral,then we get the definition of WI.Lp.log L/k; `1/.Rd/ .k 2 N/. A function f 2WI.Lp; `1/.Rd/ belongs to the space WI.Lp; c0/.Rd/ .1 � p � 1/ if for all � > 0

there exists K 2 N such that

f1.Œ�K;K�d/c

WI .Lp;`1/< �:

The space WI.Lp.log L/k; c0/.Rd/ is defined analogously.In the one-dimensional case the WI spaces are the same as the usual W spaces. It

is easy to see that

WI.Lp1 ; `1/.Rd/ WI.Lp2 ; `1/.Rd/ . p1 � p2/:

For all 1 � p � 1 and k 2 N,

W.Lp; `1/.Rd/ WI.Lp; `1/.Rd/;

W.Lp.log L/k; `1/.Rd/ WI.Lp.log L/k; `1/.Rd/:

Moreover, for 1 � p < r � 1,

WI.Lp.log L/d�1; `1/.Rd/ C0.Rd/;

WI.Lp.log L/d�1; `1/.Rd/ WI.Lr; `1/.Rd/ Lr.Rd/;

WI.Lp; `1/.Rd/ WI.Lp.log L/d�1; `1/.Rd/ Lp.log L/d�1.Rd/;

WI.Lp; `1/.Rd/ Lp.Rd/ Lp.log L/d�1.Rd/:

Definition 3.1.10 The strong maximal function is defined by

Ms f .x/ WD supx2I

1

jIjZ

Ij f j d�;

where f 2 Lloc1 .R

d/, x 2 Rd and the supremum is taken over all rectangles I D

I1 � � � Id � Rd with sides parallel to the axes.

In the one-dimensional case Ms is the usual Hardy-Littlewood maximal functionand so, it is of weak type .1; 1/. For higher dimensions it is known that there is afunction f 2 L1.Rd/ such that Ms f D 1 almost everywhere (see Jessen et al. [190]and Saks [281]). Thus Ms cannot be of weak type .1; 1/; however, with the help ofthe Lp.log L/k.Rd/ spaces, we can show a weak type inequality.


Lemma 3.1.11 A sublinear operator T is simultaneously bounded on L1.R/ andis of weak type .1; 1/ if and only if

�.jTf j > 2�/ � C0�

Z 1

�=C1

�.j f j > t/ dt .� > 0/: (3.1.3)

Proof Suppose that T is of weak type .1; 1/ and is bounded on L1.R/ with boundsC0 and C1, respectively. Let us decompose f into the sum of f0 2 L1.R/ and f1 2L1.R/. For an arbitrary � > 0, set

f1;�.t/ WD�

f .t/; if j f .t/j � �=C1I.�=C1/sign f .t/; otherwise.

and f0;�.t/ D f .t/ � f1;�.t/. Then k f1;�k1 � �=C1. Since

jTf j � jTf0;�j C jTf1;�j and

Tf1;�

1 � C1

f1;�

1 � �;

we have

fjTf j > 2�g � fjTf0;�j > �g [ fjTf1;�j > �g� fjTf0;�j > �g:

Hence

�.jTf j > 2�/ � �.jTf0;�j > �/

� C0�

f0;�

1

D C0�

Z

fj f j>�=C1g.j f j � �=C1/ d�

D C0�

Z

Rd

Z 1

0

1fj f j>t>�=C1g dt d�

D C0�

Z 1

�=C1

�.j f j > t/ dt:

Suppose that (3.1.3) holds. The weak type .1; 1/ inequality follows by replacing�=C1 by 0 in (3.1.3). If f 2 L1.R/, then �.j f j > t/ D 0 as soon as t � k f k1 andthe integral vanishes if � � C1k f k1. Thus �.jTf j > 2�/ D 0 if � � C1k f k1 andso kTf k1 � 2C1k f k1. �


Lemma 3.1.12 Let the interval I � R with length 1. If a sublinear operator T issimultaneously bounded on L1.I/ and is of weak type .1; 1/, then for every k 2 P

and f 2 L1.log L/k.I/,

jTf j �logC jTf j�k�1

L1.I/� C C C

j f j �logC j f j�k

L1.I/:

Proof Observe that

j f j �logC j f j�k�1

1DZ 1

0

�.j f j > �/d.�.logC �/k�1/d�

d�: (3.1.4)

First assume that k D 1 and notice that �.jTf j � 1/ � 1. Then

Z

fjTf j>1gjTf .t/j dt D

Z 1

0

�.jTf j > � _ 1/ d�

DZ 1

1

�.jTf j > �/ d�C �.jTf j > 1/:

Since

�.jTf j > 1/ � Ck f k1 D CZ

fj f j�egj f j d�C C

Z

fj f j>egj f j d�

� C C CZ

Ij f j logC j f j d�;

it is enough to estimate the termR11�.jTf j > �/ d�.

Let us continue the proof for arbitrary k � 1. For k > 1, we have to integrate onthe right-hand side of (3.1.4) from 1 to 1. Inequality (3.1.3) implies

Z 1

1

�.jTf j > �/d.�.logC �/k�1/d�

d�

�Z 1

1

2C0�

Z 1

�=2C1

�.j f j > t/ dtd.�.logC �/k�1/

d�d�

D 2C0

Z 1

1=2C1

�.j f j > t/Z 2C1t

1

1

�

d.�.logC �/k�1/d�

d� dt:

Since


D .logC �/k�1 C .k � 1/.logC �/k�2;

an easy calculation shows that

Z 2C1t

1

1

�


d� D 1

k.logC.2C1t//

k C .logC.2C1t//k�1

D 1

k2C1

d.2C1t.logC 2C1t/k/

dt:

Therefore

Z 1

1

�.jTf j > �/d.�.logC �/k�1/d�

d�

� CZ 1

1=2C1

�.j f j > t/d.2C1t.logC 2C1t/k/

dtdt

D Ck j2C1 f j.logC j2C1 f j/kk1� C C Ck j f j.logC j f j/kk1;

which completes the proof of the lemma. �Since the classical Hardy-Littlewood maximal operator M is bounded on L1.R/

and is of weak type .1; 1/, this lemma implies

Corollary 3.1.13 Let I � R with jIj D 1. For k 2 P and f 2 L1.log L/k.I/, we have

kMf kL1.log L/k�1.I/ � C C C k f kL1.log L/k.I/ :

Theorem 3.1.14 Let I D I1 � � � Id with jI1j D � � � D jIdj D 1. If f 2L1.log L/d�1.Rd/, then

sup�>0

��.x W Ms f .x/ > �; x 2 I/ � C C C k f kL1.log L/d�1 : (3.1.5)

Moreover, for 1 < p � 1,

kMs f kp � Cp k f kp . f 2 Lp.Rd//: (3.1.6)

Proof Let us denote the one-dimensional Hardy-Littlewood maximal function inthe ith dimension by M.i/. Then

Ms f � M.1/ ı M.2/ ı � � � ı M.d/f :

By Theorem 1.2.4 and Corollary 3.1.13,

sup�>0

��.x W Ms f .x/ > �; x 2 I/ � 3

M.2/ ı � � � ı M.d/f

L1.I/

� C C C

M.3/ ı � � � ı M.d/f

L1.log L/.I/

� : : : � C C C k f kL1.log L/d�1.I/ :

Inequality (3.1.6) can be shown in the same way. �We will generalize this theorem for larger Wiener amalgam spaces. First we

define the local strong maximal function by

ms f .x/ WD supx2IjI1j�1;:::;jId j�1

1

jIjZ

Ij f j d� .x 2 R

d/;

where f 2 Lloc1 .R

d/ and the sides of the rectangles I D I1 � � � Id are parallel tothe axes. It is easy to see that inequalities (3.1.5) and (3.1.6) imply

kms f kW.L1;1 ;`1/ � C C C k f kW.L1.log L/d�1;`1/ (3.1.7)

for f 2 W.L1.log L/d�1; `1/.Rd/ and

kms f kW.Lp ;`1/ � Cp k f kW.Lp ;`1/ . f 2 W.Lp; `1/.Rd// (3.1.8)

for 1 0

� �.j f j > �; Œk; k C 1//1=p:

For j 2 f1; : : : ; dg, let us denote the local strong maximal function taken in the

dimensions 1 � i1 < i2 < : : : < ij � d by m.i1;i2;:::;ij/s .

Theorem 3.1.15 For f 2 WI.L1.log L/d�1; `1/.Rd/,

kMs f kW.L1;1;`1/ � C C Ck f kWI .L1.log L/d�1;`1/ (3.1.9)

and for 1 < p � 1

kMs f kW.Lp ;`1/ � Cpk f kWI .Lp;`1/ . f 2 WI.Lp; `1/.Rd//: (3.1.10)

Proof If d D 1, then it is easy to see that

Ms f � ms f C C k f kW.L1;`1/ :


Similarly, if d D 2, then

Ms f � ms f C Cm.1/s

supn22Z

Z n2C1

n2

j f .�; x2/j dx2

!

C Cm.2/s

supn12Z

Z n1C1

n1

j f .x1; �/j dx1

!

C C k f kW.L1;`1/ : (3.1.11)

Taking the Lp-norm of the second summand, by (3.1.8) and Hölder’s inequality weobtain

Z n1C1

n1

ˇ

ˇ

ˇ

ˇ

ˇ

m.1/s

supn22Z

Z n2C1

n2

j f .x1; x2/j dx2

!ˇ

ˇ

ˇ

ˇ

ˇ

p

dx1

� Cp supn12Z

Z n1C1

n1

ˇ

ˇ

ˇ

ˇ

ˇ

supn22Z

Z n2C1

n2

j f .x1; x2/j dx2

ˇ

ˇ

ˇ

ˇ

ˇ

p

dx1

� Cp supn12Z

Z n1C1

n1

supn22Z

Z n2C1

n2

j f .x1; x2/jp dx2 dx1

� Cpk f kpWI .Lp;`1/:

Hence

supn1;k2Z

Z kC1

k

Z n1C1

n1

ˇ

ˇ

ˇ

ˇ

ˇ

m.1/s

supn22Z

Z n2C1

n2

j f .x1; x2/j dx2

!ˇ

ˇ

ˇ

ˇ

ˇ

p

dx1 dx2

� Cpk f kpWI .Lp;`1/:

The same holds for the third summand of (3.1.11). This and (3.1.8) prove (3.1.10).Moreover, by the one-dimensional version of (3.1.7),

sup�>0

� �

ˇ

ˇ

ˇ

ˇ

ˇ

m.1/s

supn22Z

Z n2C1

n2

j f .x1; x2/j dx2

!ˇ

ˇ

ˇ

ˇ

ˇ

> �; x1 2 Œn1; n1 C 1/

!

� C supn12Z

Z n1C1

n1

supn22Z

Z n2C1

n2

j f .x1; x2/j dx2 dx1

� Cpk f kWI .L1;`1/

and

supn1;k2Z

sup�>0

� �

ˇ

ˇ

ˇ

ˇ

ˇ

m.1/s

supn22Z

Z n2C1

n2

j f .x1; x2/j dx2

!ˇ

ˇ

ˇ

ˇ

ˇ

> �;

x 2 Œn1; n1 C 1/ Œk; k C 1/

!

� Ck f kWI .L1;`1/:


For the other terms of (3.1.11) the inequalities can be proved similarly, whichshows (3.1.9).

If d D 3, then on the right-hand side of (3.1.11), we get the terms ms f ,k f kW.L1;`1/ and some other terms similar to

m.1/s

supn2;n32Z

Z n2C1

n2

Z n3C1

n3

j f .�; x2; x3/j dx2 dx3

!

(3.1.12)

and

m.1;2/s

supn32Z

Z n3C1

n3

j f .�; �; x3/j dx3

!

: (3.1.13)

The inequalities are clear for the first two terms, i.e. for ms f and k f kW.L1;`1/.Equation (3.1.10) can be proved for the third and fourth terms, i.e. for (3.1.12)and (3.1.13), and (3.1.9) for (3.1.12) as above. We have to prove (3.1.9) for (3.1.13).By (3.1.7),

sup�>0

� �

ˇ

ˇ

ˇ

ˇ

ˇ

m.1;2/s

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j dx3

!ˇ

ˇ

ˇ

ˇ

ˇ

> �;

x1 2 Œn1; n1 C 1/; x2 2 Œn2; n2 C 1/

!

� C C C supn1;n22Z

Z n1C1

n1

Z n2C1

n2

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j dx3

!

logC

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j dx3

!

dx1 dx2:

Now let us apply Jensen’s inequality to obtain

sup�>0

� �

ˇ

ˇ

ˇ

ˇ

ˇ

m.1;2/s

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j dx3

!ˇ

ˇ

ˇ

ˇ

ˇ

> �;

x1 2 Œn1; n1 C 1/; x2 2 Œn2; n2 C 1/

!

� C C C supn1;n22Z

Z n1C1

n1

Z n2C1

n2

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j logC j f .x1; x2; x3/j dx3 dx1 dx2:

This can be estimated easily by

sup�>0

� �

ˇ

ˇ

ˇ

ˇ

ˇ

m.1;2/s

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j dx3

!ˇ

ˇ

ˇ

ˇ

ˇ

> �;

x1 2 Œn1; n1 C 1/; x2 2 Œn2; n2 C 1/

!

� C C C supn12Z

Z n1C1

n1

supn22Z

Z n2C1

n2

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j�

logC j f .x1; x2; x3/j�2

dx3 dx2 dx1

� C C Ck f kWI .L1.log L/d�1;`1/:

Hence

supn1;n2;k2Z

sup�>0

� �

ˇ

ˇ

ˇ

ˇ

ˇ

m.1;2/s

supn32Z

Z n3C1

n3

j f .x1; x2; x3/j dx3

!ˇ

ˇ

ˇ

ˇ

ˇ

> �;

x 2 Œn1; n1 C 1/ Œn2; n2 C 1/ Œk; k C 1/

!

� C C Ck f kWI .L1.log L/d�1;`1/:

This implies the inequality (3.1.9). The theorem can be proved for d > 3 in thesame way. �

We generalize the preceding results as follows. For 1 � p < 1 let

Ms;pf .x/ WD supx2I

�

1

jIjZ

Ij f jp d�

�1=p

.x 2 Rd/;

where the supremum is taken over all rectangles with sides parallel to the axes. SinceMp

s;pf D Ms.j f jp/ for 1 � p < 1, we have

sup�>0

��.x W Ms;pf .x/ > �; x 2 I/1=p � Cp C Cp k f kLp.log L/d�1

for f 2 Lp.log L/d�1.Rd/ and I D I1 � � � Id with jI1j D � � � D jIdj D 1. Moreover,

Ms;pf


d/; p < r � 1/:

Theorems 3.1.14 and 3.1.15 imply

3.2 Tempered distributions and Hardy spaces 151

Corollary 3.1.16 Assume that 1 � p < 1 and I D I1 � � � Id with jI1j D � � � DjIdj D 1. If f 2 Lp.log L/d�1.Rd/, then

sup�>0

��.x W Ms;pf .x/ > �; x 2 I/1=p � Cp C Cp k f kLp.log L/d�1 :

For p < r � 1,

Ms;pf


d//:

If f 2 WI.Lp.log L/d�1; `1/.Rd/, then

Ms;pf

W.Lp;1;`1/� Cp C Cp k f kWI .Lp.log L/d�1;`1/

and, for p < r � 1,

Ms;pf

W.Lr ;`1/� Cr k f kWI .Lr ;`1/ . f 2 WI.Lr; `1/.Rd//:

3.2 Multi-Dimensional Tempered Distributionsand Hardy Spaces

As in Chap. 1, the various characterizations of the multi-dimensional Hardy spacesare not proved here. For these characterizations see the books and papers ofDuren [93], Stein [308, 309], Stein and Weiss [311], Lu [233], Uchiyama [340],Fefferman and Stein [108], Chang and Fefferman [56, 58, 105, 107] as well asGundy and Stein [164, 165].

The Schwartz functions are defined for higher dimensions as follows.

Definition 3.2.1 The function f 2 C1.Rd/ is called a Schwartz function if for all˛; ˇ 2 N

d,

supx2Rd

ˇ

ˇx˛@ˇf .x/ˇ

ˇ D C˛;ˇ < 1:

We use the conventional notations

x˛ D x˛11 � � � x˛dd ; @ˇ D @

ˇ11 � � � @ˇd

d ;

where ˛ D .˛1; : : : ; ˛d/ and ˇ D .ˇ1; : : : ; ˇd/. The tempered distributions aredefined as in the one-dimensional case and have the same properties. The sets ofSchwartz functions and tempered distributions are denoted by S.Rd/ and S0.Rd/,respectively.

For a function � on Rd let

�t.x/ WD t�d�.x=t/ .t > 0/

and the multi-dimensional Poisson kernel let be defined by

Pd.x/ WD Pd1.x/ WD cd

.1C jxj2/.dC1/=2 ;

Pdt .x/ D t�dPd.x=t/ D cdt

.t2 C jxj2/.dC1/=2 ; .t > 0; x 2 Rd/:

The constant cd is chosen such thatR

Rd Pd.x/ dx D 1. For a bounded tempereddistribution f 2 S0.Rd/ let

P5

� f .x/ WD supt>0

supy2RdWjx�yj<t

j. f � Pdt /. y/j;

PC� f .x/ WD sup

t>0j. f � Pd

t /.x/j

and, for x D .x1; : : : ; xd/ 2 Rd,

P5 f .x/ WD supti>0;jxi�yij<ti;iD1;:::;d

j. f � .Pt1 ˝ � � � ˝ Ptd //. y/j ;

PCf .x/ WD supti>0;iD1;:::;d

j. f � .Pt1 ˝ � � � ˝ Ptd//.x/j;

where Pti WD P1ti . Let us introduce the hybrid maximal functions by

f]i.x/ WD suptk>0;kD1;:::;dIk¤i

j. f � .Pt1 ˝ � � � ˝ Pti�1 ˝ PtiC1˝ � � � ˝ Ptd //.x/j;

where xi is fixed.

Definition 3.2.2 For 0 < p < 1, the Hardy spaces H�p .R

d/, Hp.Rd/, the weak

Hardy spaces H�p;1.Rd/, Hp;1.Rd/ and the hybrid Hardy spaces Hi

p.Rd/ .i D

1; : : : ; d/ consist of all bounded tempered distributions for which

k f kH�p

WD

P5

� f

p< 1; k f kHp

WD kP5 f kp < 1;

k f kH�p;1

WD

P5

� f

p;1 < 1; k f kHp;1WD kP5 f kp;1 < 1

and

k f kHip

WD

f]i

p< 1;

respectively.

3.2 Tempered distributions and Hardy spaces 153

For p D 1 let again H�1.Rd/ WD H1.Rd/ WD L1.Rd/. For � 2 S.Rd/, �i 2S.R/ with

R

Rd � d� ¤ 0 andR

R�i d� ¤ 0 .i D 1; : : : ; d/ let

�5

� f .x/ WD supt>0

supy2RdWjx�yj<t

j. f � �t/. y/j;

�C� f .x/ WD sup

t>0j. f � �t/.x/j

and

�5 f .x/ WD supti>0;jxi�yij<ti ;iD1;:::;d

j. f � .�1;t1 ˝ � � � ˝ �d;td//. y/j ;

�Cf .x/ WD supti>0;iD1;:::;d

j. f � .�1;t1 ˝ � � � ˝ �d;td//.x/j:

Let m 2 P, N. p/ WD bd.1=p � 1/c, m > N. p/, � 2 S.R/

k�kKm WD supx2Rd ;j˛j�m

.1C jxj/mCd j@˛�.x/j

and

f�.x/ WD fm;�.x/ WD supk�kKm �1

�5

�. f /.x/:

For x 2 Rd we define the cone

.x/ WD f. y; t/ 2 Rd RC W jx � yj < tg;

where RC denotes the positive real numbers. The Lusin area integrals are intro-duced by

S� f .x/ WD S';� f .x/ WD�Z

.x/j. f � 't/. y/j2 dy dt

tdC1

�1=2

and

S f .x/ WD�Z

.x1/��.xd/

j. f � . t1 ˝ � � � ˝ td //. y/j2 dy dt

t21 � � � t2d

�1=2

;

where x 2 Rd, ' 2 C1

c .Rd/ and 2 C1

c.R/ satisfyingR

Rd ' d� D 0 andR

R d� D

0. Note that 2 C1c means that continuously differentiable with compact support.

Theorem 3.2.3 For a tempered distribution f 2 S0.Rd/ and for 0 < p < 1, wehave the following equivalences:

kP5

� f kp � kPC� f kp � k�5

� f kp � k�C� f kp � k fm;�kp � kS';� f kp

kP5 f kp � kPCf kp � k�5 f kp � k�Cf kp � kS f kp;

whereR

R� d� ¤ 0,

R

R�i d� ¤ 0 .i D 1; : : : ; d/,

R

Rd ' d� D 0,R

R d� D 0 and

m > N. p/.We can show as in the one-dimensional case that Theorems 1.4.12 and 1.4.13

hold for both the spaces H�p .R

d/ and Hp.Rd/.

3.3 Inequalities with Respect to Multi-DimensionalHardy Spaces

We show that L1.Rd/ � H�1;1.Rd/ and Hi

1.Rd/ � H1;1.Rd/ .i D 1; : : : ; d/. In

higher dimensions, the space Hi1.R

d/ will play the role of the L1.Rd/ space.

Theorem 3.3.1 We have

k f kH�1;1

D sup�>0

��.P5

� f > �/ � Ck f k1 . f 2 L1.Rd//:

For f 2 L1.log L/d�1.Rd/ and C0 > 0,

sup�>0

��.x W P5 f .x/ > �; jxj � C0/ � C C C

j f j.logC j f j/d�1

1:

If 1 < p � 1, then

k f kH�p

D

P5

� f

p � Cp k f kp . f 2 Lp.Rd//:

and

k f kHpD kP5 f kp � Cp k f kp . f 2 Lp.R

d//:

Proof The inequality P5

� f � CMf can be shown as in the proof of Theorem 1.5.1.Moreover, P5 f � CMs f follows in the same way. �

We can sharpen this result as follows.

Theorem 3.3.2 For i D 1; : : : ; d, we have

k f kH1;1 D sup�>0

��.P5 f > �/ � C k f kHi1

. f 2 Hi1.R

d// (3.3.1)

3.3 Inequalities with respect to Hardy spaces 155

and, for any C0 > 0,

f]i1B.0;C0/

1� C C C

j f j.logC j f j/d�1

1(3.3.2)

if f 2 L1.log L/d�1.Rd/.

Proof For simplicity, we show (3.3.1) for d D 2, only. By the positivity of thePoisson kernel, we have

�

x W supjyi�xij�ti ;iD1;2

ˇ

ˇ

ˇ

ˇ

Z

R2

f .u/Pt1. y1 � u1/Pt2 . y2 � u2/ du

ˇ

ˇ

ˇ

ˇ

> �

!

� �

x W supjy2�x2j�t2

Z

R

supjy1�x1j�t1

ˇ

ˇ

ˇ

ˇ

Z

R

f .u1; u2/Pt1 . y1 � u1/ du1

ˇ

ˇ

ˇ

ˇ

!

Pt2 . y2 � u2/ du2 > �

!

:

Let us denote this set by H. Applying Fubini’s theorem and the one-dimensionalinequality (1.5.1), we get that the right-hand side is equal to

Z

R

Z

R

1H.x/ dx2 dx1

� C

�

Z

R

Z

R

supjy1�x1j�t1

ˇ

ˇ

ˇ

ˇ

Z

R

f .u1; x2/Pt1 . y1 � u1/ du1

ˇ

ˇ

ˇ

ˇ

dx2 dx1

D C

�

Z

R

Z

R

f]2.x/ dx

which proves (3.3.1). By Corollary 3.1.13, inequality (3.3.2) can be shown in thesame way. �

With a similar argument as in the one-dimensional case, we can see for 1 < p �1 that the Hardy spaces coincide again with the Lp.R

d/ spaces and H1.Rd/ is a

proper space of L1.Rd/.

Theorem 3.3.3 If 1 < p < 1, then Hp.Rd/ � H�

p .Rd/ � Hi

p.Rd/ � Lp.R

d/ .i D1; : : : ; d/ and

k f kp � k f kH�p

� k f kHp� Cp k f kHi

p� Cp k f kp :

Theorem 3.3.4 For p D 1, H1.Rd/ � H�

1 .Rd/ � L1.Rd/, Hi

1.Rd/ � L1.Rd/ .i D

1; : : : ; d/ and

k f k1 � k f kH�1

� k f kH1 ; k f k1 � k f kHi1:

3.4 Atomic Decompositions

In this section we characterize the atomic decompositions of the two Hardy spacesH�

p .Rd/ and Hp.R

d/. The atomic decomposition of the first Hardy space is verysimilar to that of the one-dimensional spaces presented in Sect. 1.6 and so weomit the proofs (see Section 3.4.1). The atomic decomposition of Hp.R

d/ is morecomplicated and it will be formulated in Sect. 3.4.2.

3.4.1 Atomic Decomposition of H�p .Rd/

In the definition of the atoms we use here rather balls or cubes instead of intervals.

Definition 3.4.1 A bounded measurable function a is a ball p-atom .0 < p < 1/

if there exists a ball B � Rd such that

(i) supp a � B,(ii) kak1 � jBj�1=p,

(iii)R

B a.x/xk dx D 0 for all multi-indices k D .k1; : : : ; kd/ with jkj � N. p/ WDbd.1=p � 1/c.

Theorem 3.4.2 A tempered distribution f 2 S0.Rd/ is in H�p .R

d/ .0 < p � 1/

if and only if there exist a sequence .ak; k 2 N/ of ball p-atoms and a sequence. k; k 2 N/ of real numbers such that

1X

kD0j kjp < 1 and

1X

kD0 kak D f in S0.Rd/: (3.4.1)

Moreover,

k f kH�p

� inf

1X

kD0j kjp

!1=p

;

where the infimum is taken over all decompositions of f of the form (3.4.1).Changing (ii) in Definition 3.4.1 by

(ii) kakq � jBj1=q�1=p .0 1/,

we obtain the definition of ball . p; q/-atoms and Theorem 3.4.2 holds for theseatoms, too. In the definition of the atoms and in Theorem 3.4.2 we can also usecubes instead of balls.

Definition 3.4.3 A bounded measurable function a is a cube . p; q/-atom .0 1/ if there exists a cube I � R

d such that

3.4 Atomic Decompositions 157

(i) supp a � I,(ii) kakq � jIj1=q�1=p,

(iii)R

I a.x/xk dx D 0 for all multi-indices k D .k1; : : : ; kd/ with jkj � N. p/.

For q D 1 we call them cube p-atoms. We can prove Theorem 3.4.2 for theseatoms as well.

3.4.2 Atomic Decomposition of Hp.Rd/

The results of this subsection will be applied later in Sects. 3.5.2 and 3.6.2 and inSect. 6.4. One way to generalize the definition of the one-dimensional atoms forhigher dimensions is Definition 3.4.1, then we obtain the Hardy space H�

p .Rd/.

Another way would be the following:

(i) supp a � I, I � Rd is a rectangle,

(ii) kakq � jIj1=q�1=p,(iii)

R

Ra.x/xk dxi D 0, for all i D 1; : : : ; d.

However, the space Hp.Rd/ do not have atomic decomposition with respect to

these atoms (see Weisz [347]). The atomic decomposition for Hp.Rd/ is much more

complicated. One reason of this is that the support of an atom is not a rectangle butan open set. Moreover, here we have to choose the atoms from L2.Rd/ instead ofL1.Rd/.

First of all we introduce some notations. By a dyadic interval we mean one ofthe form Œk2�n; .k C 1/2�n/ for some k; n 2 Z. A dyadic rectangle is the Cartesianproduct of d dyadic intervals. Suppose that F � R

d is an open set. Let M1.F/denote those dyadic rectangles R D I S � F, I � R is a dyadic interval, S �R

d�1 is a dyadic rectangle, that are maximal in the first direction. In other words, ifI0 S R is a dyadic subrectangle of F (where I0 � R is a dyadic interval) thenI D I0. Define Mi.F/ similarly. Denote by M.F/ the maximal dyadic subrectanglesof F in the above sense.

Recall that if I � R is an interval, then rI is the interval with the same centreas I and with length rjIj .r 2 N/. For a rectangle R D I1 : : : Id � R

d letrR WD rI1 : : : rId. Instead of 2rR we write Rr .r 2 N/.

Definition 3.4.4 A function a 2 L2.Rd/ is a p-atom or Hp-atom .0 < p � 1/ if

(i) supp a � F for some open set F � Rd with finite measure,

(ii) kak2 � jFj1=2�1=p,(iii) a can be decomposed further into the sum of “elementary particles” aR 2

L1.Rd/, a D P

R aR, where R � F are dyadic rectangles, such that


(a) supp aR � 5R,(b) for all R, i D 1; : : : ; d and almost every fixed x1; : : : ; xi�1; xiC1; : : : ; xd,

we haveZ

R

aR.x/xki dxi D 0 .k D 0; : : : ;M. p//;

where M. p/ WD b2=p � 3=2c,(c) aR 2 CM. p/C1 such that kaRk1 � dR for some dR > 0 and

@k11 � � � @kd

d aR

1 � dR

jI1jk1 � � � jIdjkd

for all 0 � ki � M. p/C 1 .i D 1; : : : ; d/ with

X

R�F

d2RjRj � CpjFj1�2=p;

where R D I1 � � � Id.

Moreover, a can also be decomposed into the sum of “elementary particles”˛R 2 L2.Rd/,

a DX

R2M.F.1//

˛R;

satisfying

(d) supp ˛R � 5R,(e) for all R 2 M.F.1//, i D 1; : : : ; d and almost every fixed

x1; : : : ; xi�1; xiC1; : : : ; xd,Z

R

˛R.x/xki dxi D 0 .k D 0; : : : ;M. p//;

(f) for every disjoint partition Pl .l 2 P/ of M.F.1//,

0

B

@

X

l2P

X

R2Pl

˛R

2

2

1

C

A

1=2

� jFj1=2�1=p;

where F.1/ WD fMs.1F/ > 1=100g.

In order to prove the atomic decomposition for the Hp.Rd/ spaces, we need some

lemmas and notations. For two different dyadic intervals I and IR, let

m.I; IR/ WD min.jIRj; jIj/max.jIRj; jIj/ ;

while for two different two-dimensional dyadic rectangles S D IJ and R D IRJR,let

m.S;R/ WD min.jIRj; jIj/min.jJRj; jJj/max.jIRj; jIj/max.jJRj; jJj/ :

For x 2 Rd, Sx denotes the set of the dyadic rectangles of Rd containing x.

Lemma 3.4.5 For x 2 R, IR � R dyadic interval and t > 0, we have

X

I2Sx; 5IR\5I¤;m.I; IR/

t � CMt.15IR/.x/: (3.4.2)

Proof First assume that x 2 10IR. If jIj D 2mjIRj for some m 2 Z, then

X


t �0X

mD�12mt C

1X

mD12�mt � C:

Let K � R be an interval such that x is the centre of K and jKj D 6jIRj. In this casejK \ 5IRj � jIRj=2. Thus

M.15IR/.x/ � jK \ 5IRjjKj � 1

12;

which proves (3.4.2) in case x 2 10IR.Suppose now that x 62 10IR. Then jIj � jIRj and %.x; 5IR/ � 2jIRj. Let jIRj D

2�M .M 2 Z/ and define N 2 Z such that 2�N � %.x; 5IR/ < 2�NC1. Then 2�N �2 � 2�M, or N � M � 1. If jIj D 2mjIRj D 2m2�M for some m 2 N, then

2�N � %.x; 5IR/ � 5jIj � 2m�MC3;

in other words, m � M � N � 3. Henceforth,

X


t �1X

mDM�N�32�mt � C2�.M�N/t:

Let K � R be an interval such that x is the centre of K and jKj D %.x; 5IR/C jIRj.In this case jK \ 5IRj D jIRj. Thus

M.15IR/.x/ � jK \ 5IRjjKj � 2�M

2�NC1 C 2�M� 2

32N�M

and (3.4.2) is proved. �

Lemma 3.4.6 If

F.1/ D fMs.1F/ > 1=100g ;

then for all R � F � R2, 10R � F.1/ and

ˇ

ˇF.1/ˇ

ˇ � CjFj.Proof By (3.1.6),

ˇ

ˇF.1/ˇ

ˇ � CZ

R2

Ms.1F/2 d� � C

Z

R2

12F d� D CjFj: (3.4.3)

Since x 2 10R implies

Ms.1F/.x/ >j10R \ Fj

j10Rj � jR \ Fjj10Rj D 1

100;

we have 10R � F.1/. �

Lemma 3.4.7 For x 2 R2 and R D IR JR � F � R

2 dyadic rectangles and t > 0,we have

X

S2Sx; 5R\5S¤;m.S;R/t � CMt

s.1F.1//.x/:

Proof For S D I J, Lemma 3.4.5 implies

X

S2Sx; 5R\5S¤;m.S;R/t �

X

I2Sx1 ; 5IR\5I¤;m.I; IR/

tX

J2Sx2 ; 5JR\5J¤;m.J; JR/

t

� CMt.15IR/.x1/Mt.15JR/.x2/

D CMts.15R/.x/

� CMts.1F.1/ /.x/;

which shows the lemma. �

Theorem 3.4.8 Let 0 < p � 1, 2 CM. p/C10 .R/ such that is even, supported on

Œ�1; 1� and

Z

R

.x/xk dx D 0 .0 � k � M. p//: (3.4.4)

Then for all p-atom a,

S a

p� Cp:

Proof For the sake of simplicity, we suppose that d D 2. For higher dimensionswe can show the statement in the same way. If t D .t1; t2/ 2 R

2C and x D .x1; x2/,we set

t.x/ WD t1 .x1/ t2 .x2/ D .t1t2/�1 .x1=t1/ .x2=t2/:

For a dyadic rectangle S D I J, let

SC WDn

. y; t/ 2 R2 R

2C W y 2 S; jIj < t1 � 2jIj; jJj < t2 � 2jJjo

:


S2 a.x/ DZ

.x1/.x2/ja � t. y/j2 dy dt

t21t22

�X

S2Sx

Z

SC

ja � t. y/j2 dy dt

t21t22

: (3.4.5)

Fix a point x for which

Ms.1F.1//.x/ < 1=2:

Fix S 2 Sx, S D IJ and suppose that R � F is a dyadic rectangle with 5R\5S ¤ ;,R D IR JR. Then there are four types of such dyadic rectangles R.

(i) jIRj � jIj, jJRj � jJj.This cannot occur. It is easy to see that in this case S � 10R. Hence, by Lemma 3.4.6,

1 D jS \ 10RjjSj � jS \ F.1/j

jSj � Ms.1F.1//.x/ <1

2;

which is a contradiction.

(ii) jIRj � jIj, jJRj � jJj.For fixed . y; t/ 2 SC, we have that the support of t1 . y1 � �/ is contained in 5I andthe support of t2 . y2 � �/ is contained in 5J. As in (iii) of Definition 3.4.4, let usdecompose a into the sum

a DX

R�F

aR;


where the open set F is the support of a. Then, by Taylor’s formula, (3.4.4) and (a),(b) of Definition 3.4.4, we conclude that

jaR � t. y/j Dˇ

ˇ

ˇ

ˇ

Z

R2

aR.u/ t1 . y1 � u1/ t2 . y2 � u2/ du

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

ˇ

Z

5IR

Z

5J

aR.u1; u2/�kX

lD0

@l2aR.u1; cJ/

lŠ.u2 � cJ/

l

!

t1 . y1 � u1/�kX

lD0

.l/t1 . y1 � cIR/

lŠ.cIR � u1/

l

!

t2 . y2 � u2/ du1 du2

ˇ

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

ˇ

Z

5IR

Z

5J

@kC12 aR.u1; /

.k C 1/Š.u2 � cJ/

kC1

.kC1/t1 . y1 � �/.k C 1/Š

.cIR � u1/kC1 t2 . y2 � u2/ du1 du2

ˇ

ˇ

ˇ

ˇ

ˇ

;

where 0 � k � M. p/, cIR and cJ are the centres of IR and J, respectively, 2.cJ; u2/ and � 2 .cIR ; u1/. Since

.kC1/t1 .x1/ D t�k�2

1 .kC1/.x1=t1/;

(c) implies

jaR � t. y/j � C

�

dR

jJRjkC1 jJjkC1 jIRjkC1

jIjkC21

jJj�

jIRjjJj

� C

� jIRjjJjjIjjJRj

�kC1dR

jIRjjIj

D Cm.S;R/kC1dRjIRjjIj :

Notice that C is depending on k .kC1/k1. Since jIRj � jIj and jJRj � jJj, we have

jIRjjIj � 25

j5R \ 5Sjj5Sj D 25

j5SjZ

5S15R.t/ dt � CMs.15R/.x/:


This implies that

jaR � t. y/j � Cm.S;R/kC1dR

� jIRjjIj�1=r � jIRj

jIj�1�1=r

(3.4.6)

� Cm.S;R/kC1dRM1=rs .15R/.x/

� jIRjjIj�1�1=r

;

where r > 1 and . y; t/ 2 SC.

(iii) jIRj � jIj, jJRj � jJj.Similar to the case (ii), we get that

jaR � t. y/j � Cm.S;R/kC1dRM1=rs .15R/.x/

jJRjjJj

�1�1=r

for . y; t/ 2 SC and r > 1.

(iv) jIRj � jIj, jJRj � jJj.As in case (ii), we obtain for . y; t/ 2 SC that

jaR � t. y/j Dˇ

ˇ

ˇ

ˇ

Z

R2

aR.u/ t1 . y1 � u1/ t2 . y2 � u2/ du

ˇ

ˇ

ˇ

ˇ

Dˇ

ˇ

ˇ

ˇ

ˇ

Z

5IR

Z

5JR

aR.u1; u2/

t1 . y1 � u1/ �kX

lD0

.l/t1 . y1 � cIR/

lŠ.cIR � u1/

l

!

t2 . y2 � u2/ �kX

lD0

.l/t2 . y2 � cJR/

lŠ.cJR � u2/

l

!

du1 du2

ˇ

ˇ

ˇ

ˇ

ˇ

;

where 0 � k � M. p/, cIR and cJR are the centres of IR and JR, respectively. UsingTaylor’s formula, we see

jaR � t. y/j Dˇ

ˇ

ˇ

ˇ

ˇ

Z

5IR

Z

5JR

aR.u1; u2/ .kC1/t1 . y1 � /.k C 1/Š

.cIR � u1/kC1

.kC1/t2 . y2 � �/

.k C 1/Š.cJR � u2/

kC1 du1 du2

ˇ

ˇ

ˇ

ˇ

ˇ

� C

�Z

5IR

Z

5JR

jaR.u1; u2/j du1 du2

� jIRjkC1

jIjkC2jJRjkC1

jJjkC2


for suitable 2 .cIR ; u1/, � 2 .cJR ; u2/. If r > 1 and 1=r C 1=r0 D 1, then byHölder’s inequality,

jaR � t. y/j � C

�

1

jIjjJjZ

R2

jaR.u/jr du

�1=r

(3.4.7)

�

1

jIjjJjZ

R2

15R.u/ du

�1=r0 � jIRjjJRjjIjjJj

�kC1

� CM1=rs .ar

R/.x/

� jIRjjJRjjIjjJj

�kC1C1=r0

� CdRM1=rs .15R/.x/m.S;R/

kC2�1=r:

For . y; t/ 2 SC we have

ja � t. y/j2 Dˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R�F; 5R\5S¤;aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

(3.4.8)

� C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .ii/

aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

C C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .iii/

aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

C C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .iv/

aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

:

Applying Cauchy-Schwartz inequality and (3.4.6), we can see thatˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .ii/

aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

� C

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .ii/

m.S;R/k=2C1�1=2rdRM1=rs .15R/.x/

m.S;R/k=2C1=2r


ˇ

ˇ

ˇ

ˇ

ˇ

2

� CX

R2Type .ii/

d2RM2=rs .15R/.x/m.S;R/

kC2�1=r

X

R2Type .ii/

m.S;R/kC1=r


� CX

R�F; 5R\5S¤;d2RM2=r

s .15R/.x/m.S;R/kC2�1=r

X

R2Type .ii/

jIRjjIj�kC2�1=r jJj

jJRj�kC1=r

:


Fix S 2 Sx. If R 2 Type .ii/, then jIRj D 2�mjIj and jJRj D 2njJj for some m; n 2 N.Since 5IR \ 5I ¤ ;, IR � 10I. If m 2 N is fixed, then we have at most 10 � 2m suchIR. Similarly, for a fixed n 2 N we have at most 10 such JR. Thus for any t > 1,u > 0,

X

R2Type .ii/

� jIRjjIj�t � jJj

jJRj�u

� CX

m2N

X

n2N2m2�mt2�nu � C:

Hence

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .ii/

aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

� CX


s .15R/.x/m.S;R/kC2�1=r:

Similarly,

ˇ

ˇ

ˇ

X

R2Type .iii/

aR � t. y/ˇ

ˇ

ˇ

2

� CX


s .15R/.x/m.S;R/kC2�1=r:

By (3.4.7),

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .iv/

aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

� C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .iv/

m.S;R/.kC2�1=r/=2dRM1=rs .15R/.x/m.S;R/

.kC2�1=r/=2

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

� CX


s .15R/.x/m.S;R/kC2�1=r

X

R2Type .iv/

m.S;R/kC2�1=r:

Similarly as above, for any t > 1, we obtain

X

R2Type .iv/

m.S;R/t DX

R2Type .iv/

� jIRjjJRjjIjjJj

�t

� CX

m2N

X

n2N2m2�mt2n2�nt � C

and so

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2Type .iv/

aR � t. y/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

2

(3.4.9)

� CX


s .15R/.x/m.S;R/kC2�1=r:

Combining the inequalities (3.4.8)–(3.4.9), we conclude

ja � t. y/j2 � CX


s .15R/.x/m.S;R/kC2�1=r:

For r > 1 and for a point x with Ms.1F.1//.x/ < 1=2, we get by Lemma 3.4.7and (3.4.5) that

S2 a.x/ �X

S2Sx

Z

SC

ja � t. y/j2 dy dt

t21t22

� CX

S2Sx

X


s .15R/.x/m.S;R/kC2�1=r

� CX

R�F

0

@

X

S2Sx; 5R\5S¤;m.S;R/kC2�1=r

1

A d2RM2=rs .15R/.x/

� CMkC2�1=rs .1F.1//.x/

X

R�F

d2RM2=rs .15R/.x/:

Using Hölder’s inequality, we get that

Z

fMs.1F.1//.x/<1=2g

Sp a.x/ dx

� CZ

R2

Mp.kC2�1=r/=2s .1F.1//.x/

X

R�F

d2RM2=rs .15R/.x/

!p=2

dx

� C

�Z

R2

Mp.kC2�1=r/=.2�p/s .1F.1//.x/

�1�p=2

Z

R2

X

R�F

d2RM2=rs .15R/.x/ dx

!p=2

:

Choose r and k such that 1 < r < 2 and p.k C 2 � 1=r/=.2 � p/ > 1. By an easycalculation, we can see that the smallest such k is b2=p � 3=2c. Then

Z

fMs.1F.1/ /.x/<1=2gSp a.x/ dx

� C

�Z

R2

Mp.kC2�1=r/=.2�p/s .1F.1//.x/

�1�p=2

X

R�F

d2RjRj!p=2

� CjF.1/j1�p=2jFjp=2�1 � C;

because of (3.4.3). On the other hand,Z

fMs.1F.1//.x/�1=2g

Sp a.x/ dx

� C

�Z

R2

S2 a.x/ dx

�p=2

jfMs.1F.1//.x/ � 1=2gj1�p=2

� C

�Z

R2

a2.x/ dx

�p=2 �Z

R2

M2s .1F.1//.x/ dx

�1�p=2

� CjFjp=2�1jF.1/j1�p=2 � C;

which finishes the proof of the theorem. �

Lemma 3.4.9 Suppose that satisfies the conditions of Theorem 3.4.8. If f 2L2.R2/, then for almost every x 2 R

2,

�Z 1

0

ˇ

ˇ

ˇ

b .�/ˇ

ˇ

ˇ

2 d�

�

�2

f .x/ DZ

R2

Z

R2C

f � t. y/ t.x � y/dy dt

t1t2: (3.4.10)

Proof Since jb j2 is integrable,Z 1

1

ˇ

ˇ

ˇ

b .�/ˇ

ˇ

ˇ

2 d�

�< 1:

By an easy calculation,

ˇ

ˇ

ˇ

b .�/ˇ

ˇ

ˇ

2 D 1

2�

ˇ

ˇ

ˇ

ˇ

Z 1

�1 .u/.e�{�u � 1/ du

ˇ

ˇ

ˇ

ˇ

2

� 1

2�

�Z 1

�1j .u/jj�uj du

�2

D 1

2�

�Z 1

�1j .u/jjuj du

�2

j�j2;

and so

Z 1

0

ˇ

ˇ

ˇ

b .�/ˇ

ˇ

ˇ

2 d�

�< 1:

ThusZ 1

0

ˇ

ˇ

ˇ

b .�/ˇ

ˇ

ˇ

2 d�

�< 1:

The right-hand side of (3.4.10) is equal to

Z

R2C

. f � t � t/.x/dt

t1t2:

Let us take the Fourier transform of this expression:

Z

R2C

. f � t � t/^.�/

dt

t1t2DZ

R2C

bf .�/b t.�/b t.�/dt

t1t2

Dbf .�/Z

R2C

ˇ

ˇ

ˇ

b t1 .�1/ˇ

ˇ

ˇ

2 ˇˇ

ˇ

b t2 .�2/ˇ

ˇ

ˇ

2 dt

t1t2

Dbf .�/Z

RC

ˇ

ˇ

ˇ

b .t1�1/ˇ

ˇ

ˇ

2 dt1t1

Z

RC

ˇ

ˇ

ˇ

b .t2�2/ˇ

ˇ

ˇ

2 dt2t2:

Note that the Fourier transform of two-dimensional functions will be defined laterin the next chapter. If the Fourier transforms of two functions are the same, thenthe two functions are almost everywhere equal, which implies the equality of thelemma. �

Theorem 3.4.10 A tempered distribution f 2 S0.Rd/ is in Hp.Rd/ .0 < p � 1/ if

and only if there exist a sequence .ak; k 2 N/ of p-atoms and a sequence . k; k 2 N/

of real numbers such that

1X

kD0j kjp < 1 and

1X

kD0 kak D f in S0.Rd/: (3.4.11)

Moreover,

k f kHp� inf

1X

kD0j kjp

!1=p

;

where the infimum is taken over all decompositions of f of the form (3.4.11).

Proof In this proof we are using the Lusin area function S . Recall that the Hardyspace is independent of the choice of . Again, it is enough to suppose that d D 2.Theorems 3.2.3 and 3.4.8 imply that

kakHp� Cp

S a

p � Cp;

where is chosen in Theorem 3.4.8. If f 2 S 0.R2/ has a decompositionsatisfying (3.4.11), then

Z

R2

Sp f d� �

1X

kD0j kjp

Z

R2

Sp ak d� � Cp

1X

kD0j kjp

and so f 2 Hp.R2/. Note that in this part of the proof we did not use the conditions

(d), (e) and (f) of Definition 3.4.4.Now suppose that f 2 Hp.R

2/\ L2.R2/. Recall that L2.R2/\ Hp.R2/ is dense in

Hp.R2/. As we have shown in Lemma 3.4.9, we can normalize such that

Z 1

0

ˇ

ˇ

ˇ

b .�/ˇ

ˇ

ˇ

2 d�

�D 1:

If for each dyadic rectangle R,

fR.x/ WDZ

RC


t1t2; (3.4.12)

then we get immediately from Lemma 3.4.9 that

f .x/ DX

R

fR.x/:

Set

Fk WD fS f > 2kg and F.1/k WD fMs.1Fk/ > 1=100g .k 2 Z/

and consider the collection Rk of all dyadic rectangles for which

jR \ FkC1j < 1

2jRj and jR \ Fkj � 1

2jRj:

If R is dyadic then it is easy to see that R 2 Rk for exactly one k 2 Z. For R 2 Rk

let

aR WD A

2kjFkj1=pfR; ak WD

X

R2Rk

aR

and

k WD 2kjFkj1=p

A;

where A > 0 is a constant chosen later. Then

f DX

k2Z kak

and

X

k2Zj kjp D Cp

X

k2Z2kpˇ

ˇfS f > 2kgˇˇ

D Cp

X

k2Z.2p/k

ˇ

ˇ

ˇ

n

.2p/k < Sp f � .2p/kC1

oˇ

ˇ

ˇ

� Cp

S f

p

p:

Let us show that ak is an atom with respect to F.1/k . For fixed . y; t/ 2 RC, R 2 Rk,

the support of t.x�y/ is contained in 5R. It is easy to show that 5R � F.1/k . Indeed,if x 2 5R for some R 2 Rk, then

Ms.1Fk/.x/ � j5R \ Fkjj5Rj � jR \ Fkj

j5Rj � 1

10:

Thus supp aR � 5R and supp ak � F.1/k .Now we are going to verify (ii) of Definition 3.4.4. Assume that kgk2 D 1 and

observe that the rectangles RC are disjoint. Then

ˇ

ˇ

ˇ

ˇ

Z

R2

ak.x/g.x/ dx

ˇ

ˇ

ˇ

ˇ

(3.4.13)

D A

2kjFkj1=p

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R2

X

R2Rk

Z

RC


t1t2g.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� A

2kjFkj1=p

X

R2Rk

Z

RC

j f � t. y/j jg � t. y/j dy dt

t1t2

� A

2kjFkj1=p

�Z

Uk

j f � t. y/j2 dy dt

t1t2

�1=2 �Z

Uk

jg � t. y/j2 dy dt

t1t2

�1=2

;

where Uk WD P

R2RkRC. Obviously,

1 DZ

R2

jg.x/j2 dx (3.4.14)

� CZ

R2

jS g.x/j2 dx

D CZ

R2

Z

.x1/.x2/jg � t. y/j2 dy dt

t21t22

dx

D CZ

R4

Z

R2

1fjx1�y1j<t1g1fjx2�y2j<t2g dx jg � t. y/j2 dy dt

t21t22

D CZ

R4


t1t2

� CZ

Uk


t1t2:

To estimateR

Ukj f � t. y/j2 dy dt

t1 t2, we establish that

Z

F.1/k nFkC1

ˇ

ˇS f .x/ˇ

ˇ

2dx � 22.kC1/jF.1/k j � C22kjFkj: (3.4.15)

On the other hand,

Z

F.1/k nFkC1

ˇ

ˇS f .x/ˇ

ˇ

2dx (3.4.16)

DZ

F.1/k nFkC1

Z

.x1/.x2/j f � t. y/j2 dy dt

t21t22

dx

DZ

R4

Z

F.1/k nFkC1

1fjx1�y1j<t1g1fjx2�y2j<t2g dx j f � t. y/j2 dy dt

t21t22

DZ

R4

j f � t. y/j2 �k. y; t/dy dt

t21t22

�Z

Uk


t21t22

DX

R2Rk

Z

RC


t21t22

;

where

�k. y; t/ WDˇ

ˇ

ˇ

n

x W jx1 � y1j < t1; jx2 � y2j < t2; x 2 F.1/k n FkC1oˇ

ˇ

ˇ :

Suppose . y; t/ 2 RC, R 2 Rk, and investigate �k. y; t/. Since R � F.1/k , we have

�k. y; t/ � jfx 2 R W jx1 � y1j < t1; jx2 � y2j < t2; x 62 FkC1gjD ˇ

ˇfx 2 R W jx1 � y1j < t1; jx2 � y2j < t2g \ FckC1ˇ

ˇ :

By the definition of RC,

fx 2 R W jx1 � y1j < t1; jx2 � y2j < t2g D R:

Then

�k. y; t/ � ˇ

ˇR \ FckC1ˇ

ˇ D jRj � jR \ FkC1j > 1

2jRj D 1

8t1t2:

This and inequalities (3.4.15) and (3.4.16) imply

C22kjFkj �Z

F.1/k nFkC1

ˇ

ˇS f .x/ˇ

ˇ

2dx (3.4.17)

�X

R2Rk

Z

RC


t1t2

DZ

Uk


t1t2:

Taking the supremum in (3.4.13) and considering the inequalities (3.4.14)and (3.4.17), we obtain

kakk2 D supkgk2D1

ˇ

ˇ

ˇ

ˇ

Z

R2

ak.x/g.x/ dx

ˇ

ˇ

ˇ

ˇ

� A

2kjFkj1=pCjFkj1=22k � jFkj1=2�1=p

if we choose A D 1=C.Condition (b) of Definition 3.4.4 follows easily from (3.4.4) and (3.4.12).

Observe that for R 2 Rk,

ˇ

ˇ

ˇ@k11 @

k22 aR.x/

ˇ

ˇ

ˇ D A

2k jFkj1=p

ˇ

ˇ

ˇ

ˇ

ˇ

Z

RC

f � t. y/@k11 @

k22 t.x � y/

dy dt

t1t2

ˇ

ˇ

ˇ

ˇ

ˇ

� A

2k jFkj1=p

Z

RC


t1t2

!1=2

Z

RC

ˇ

ˇ

ˇ@k11 @

k22 t.x � y/

ˇ

ˇ

ˇ

2 dy dt

t1t2

!1=2

:



ˇ

ˇ

ˇ@k11 @

k22 t.x � y/

ˇ

ˇ

ˇ � C

jI1jk1 jI2jk2 jRj ;

where . y; t/ 2 RC, R D I1 I2 and 0 � ki � M. p/ C 1 .i D 1; 2/. From this itfollows that

Z

RC

ˇ

ˇ

ˇ@k11 @

k22 t.x � y/

ˇ

ˇ

ˇ

2 dy dt

t1t2

!1=2

� C

jI1jk1 jI2jk2 jRj1=2

and

@k11 @

k22 aR

1 �C2�kjFkj�1=pjRj�1=2

R

RC

j f � t. y/j2 dy dtt1t2

�1=2

jI1jk1 jI2jk2

DW dR

jI1jk1 jI2jk2:

Then

X

R2Rk

d2RjRj � C2�2kjFkj�2=pX

R2Rk

Z

RC


t1t2

D C2�2kjFkj�2=pZ

Uk


t1t2

� CpjFkj1�2=p;

which proves (c).To each R 2 Rk we associate a maximal dyadic subrectanglebR 2 M.F.1/k / such

that R �bR. For S 2 M.F.1/k / let

˛S WDX

bRDS

aR:

Obviously,

ak DX

R2M.F.1/k /

˛R

and (d) and (e) hold.


To prove (f) let Pl .l D 1; 2; : : :/ be a disjoint partition of M.F.1/k /. We will showthat

X

S2Pl

˛S

2

2

� C2�2k jFkj�2=pX

S2Pl

X

bRDS

Z

RC


t1t2: (3.4.18)

We use again duality. If kgk2 D 1, then

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R2

X

S2Pl

˛S.x/g.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

D A

2k jFkj1=p

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R2

X

S2Pl

X

bRDS

Z

RC


t1t2g.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� A

2k jFkj1=p

X

S2Pl

X

bRDS

Z

RC

j f � t. y/j jg � t. y/j dy dt

t1t2

� A

2k jFkj1=p

0

@

Z

[bRDS;S2Pl

RC


t1t2

1

A

1=2

0

@

Z

[bRDS;S2Pl

RC


t1t2

1

A

1=2

:

Taking the supremum over all g’s with kgk2 D 1 and using (3.4.14), we obtain

X

S2Pl

˛S

2

2

� C2�2k jFkj�2=pZ

[bRDS;S2Pl

RC


t1t2;

which shows (3.4.18). By (3.4.17),

X

l

X

S2Pl

˛S

2

2

� C2�2k jFkj�2=pZ

Uk


t1t2

� C jFkj1�2=p

3.5 Interpolation between Hardy spaces 175

and this finishes the proof of (f). For arbitrary f 2 Hp.Rd/ the statements of the

theorem can be proved in the same way as in Theorem 1.6.10. �This theorem was proved by Chang and Fefferman [56–58, 105, 107] in a slightly

different form (see also Weisz [355]). Using the operator P5, Wilson [383] gaveanother version of this atomic decomposition. Of course, with the preceding proofwe get an atomic decomposition of Hp.R/ D H�

p .R/; however, then we should takein Definition 1.6.9 the 2-norm instead of the 1-norm. As for the one-dimensionalHp.R/ spaces, we could suppose that the integrals in (iii) of Definition 3.4.1 (resp.(b) and (e) of Definition 3.4.4) are zero for all k for which jkj � N (resp. k � M),where N � N. p/ and M � M. p/.

3.5 Interpolation Between Multi-Dimensional Hardy Spaces

Here we present the analogous results to those in Sect. 1.7 for the Hardy spaces onR

d. Since the proofs are all similar for H�p .R

d/, we omit them. However, for thesake of completeness, we formulate the analogues of all the results in Sect. 1.7.

3.5.1 Interpolation Between the H�p .Rd/ Spaces

It is easy to see that the analogous results to Theorem 1.7.12 and to Corollary 1.7.13regarding to the Lp.R/ spaces holds for higher dimensions, too. In the proof of thenext results, we can use again the atomic decomposition.

Theorem 3.5.1 Let f 2 H�p .R

d/, y > 0 and fix 0 ygf p� d�

!1=p

:

Theorem 3.5.2 If 0 < � < 1, 0 < p0 � 1 and 0 < q � 1, then

H�p0;H�1

�

�;qD H�

p;q;1

pD 1 � �

p0:

Corollary 3.5.3 Suppose that 0 < � < 1 and 0 < p0; p1; q0; q1; q � 1. If p0 ¤ p1,then

H�p0;q0 ;H

�p1;q1

�

�;qD H�

p;q;1

pD 1 � �

p0C �

p1:

In a special case,

H�p0;H�

p1

�

�;pD H�

p ;1

pD 1 � �

p0C �

p1

and, for 1 < p1 � 1,

H�1 ;Lp1

�

�;pD Lp;

1

pD 1 � �C �

p1:

Corollary 3.5.4 If a sublinear or linear operator V is bounded from H�p0 .R

d/ to

Lp0 .Rd/ (resp. to H�

p0 .Rd/) and from Lp1 .R

d/ to Lp1 .Rd/ . p0 � 1 < p1 � 1/, then

it is also bounded from H�p;q.R

d/ to Lp;q.Rd/ (resp. to H�

p;q.Rd/) for each p0 0 and fix 0 y�ˇ

ˇ

1=2

and

khkHp� Cp

Z

fS f>ygS f p d�

!1=p

;

where S is the Lusin area integral defined before Theorem 3.2.3.

Proof Let N 2 Z and z 2 .1=2; 1� such that y D z2N . We may suppose that d D 2

and use the notations and ideas of Theorem 3.4.10. We know that

ak DX

S2M.F.1/k /

˛S DX

S2M.F.1/k /

X

bRDS

aR:


Set

g WDNX

kD�1 kak

and

h WD1X

kDNC1 kak:

The inequality

khkpHp

� Cp

Z

fS f>z2NC1gS f p d� � Cp

Z

fS f>ygS f p d�

can be shown as in Theorem 3.4.10.To prove the inequality for g, we first show that

kgk22 D

NX

kD�1 k

X

S2M.F.1/k /

X

bRDS

aR

2

2

� CNX

kD�1

X

S2M.F.1/k /

X

bRDS

Z

RC


t1t2: (3.5.1)

Assume that k�k2 D 1. Then

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R2

NX

kD�1 k

X

S2M.F.1/k /

X

bRDS

aR.x/�.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

D

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R2

NX

kD�1

X

S2M.F.1/k /

X

bRDS

Z

RC


t1t2�.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�NX

kD�1

X

S2M.F.1/k /

X

bRDS

Z

RC

j f � t. y/j j� � t. y/j dy dt

t1t2:

By Hölder’s inequality,

kgk22 D supk�k2�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z

R2

NX

kD�1 k

X

S2M.F.1/k /

X

bRDS

aR.x/�.x/ dx

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�0

@

Z

[bRDS;S2M.F

.1/k /;k�N

RC


t1t2

1

A

1=2

supk�k2�1

0

@

Z

[bRDS;S2M.F

.1/k /;k�N

RC

j� � t. y/j2 dy dt

t1t2

1

A

1=2

:

Inequality (3.4.14) implies

kgk22 � CZ

[bRDS;S2M.F

.1/k /;k�N

RC


t1t2

which shows (3.5.1). By (3.4.17),

kgk22 � CNX

kD�1

Z

Uk


t1t2

� CNX

kD�1.z2k/2 jFkj :

The proof can be completed as in Theorem 3.4.10. �Before characterizing the interpolation spaces between the Hp.R

d/ spaces weneed the following lemma.

Lemma 3.5.6 If f � 0 is a non-increasing function on .0;1/, 0 < u < 1 and0 < r � 2, then

�Z 1

u

�

x1=2f .x/�2 dx

x

�1=2

� C

�Z 1

u=4

�

x1=2f .x/�r dx

x

�1=r

: (3.5.2)

Proof Choose N 2 Z such that 2N � u < 2NC1. Then

�Z 1

u

�

x1=2f .x/�2 dx

x

�r=2

�

0X

kDNC1f .2k�1/2

Z 2k

2k�1

dx

!r=2

�0X

kDNC1f .2k�1/r2.k�1/r=2

� C0X

kDNC1

Z 2k�1

2k�2

xr=2�1f .x/r dx

� CZ 1

u=4

�

x1=2f .x/�r dx

x:

This completes the proof of (3.5.2). �

Theorem 3.5.7 If 0 < p0 � 1, 0 < � < 1 and 0 < q � 1, then

�

Hp0 ;L2�

�;qD Hp;q;

1

pD 1 � �

p0C �

2:

Proof Let f 2 Hp;q.Rd/. Denote by QS the non-increasing rearrangement of S D

S f . Let 1=˛ D 1=p0 � 1=2 and, for a fixed t, y D QS .t˛/. The correspondingtempered distributions in Theorem 3.5.5 are denoted by ht and gt. Clearly

K.t; f ;Hp0 ;L2/ � khtkHp0C t kgtk2 :

The preceding theorem implies

khtkHp0� C

Z

fS >QS .t˛/gSp0 d�

!1=p0

D C

Z t˛

0

QS .x/p0 dx

!1=p0

:

Thus, for 0 < q < 1,

Z 1

0

t�� khtkHp0

�q dt

t� C

Z 1

0

t��q

Z t˛

0

QS .x/p0 dx

!q=p0dt

t

� CZ 1

0

t.1��/q=p0C�q=2

�

1

t

Z t

0

QS .x/p0 dx

�q=p0 dt

t:

Using (1.7.7) we obtain

Z 1

0

t�� khtkHp0

�q dt

t� C

Z 1

0

t.1��/q=p0C�q=2 QS .t/q dt

tD C

S f

q

p;q:


On the other hand, by Theorem 3.5.5,

kgtk2 � C

Z

fS �QS .t˛/gS2 d�

!1=2

C C QS .t˛/ˇ

ˇ

˚

S > QS .t˛/�ˇ

ˇ

1=2:

Since the distributions of S and QS are identical and QS is non-increasing,

ˇ

ˇ

˚

S > QS .t˛/�ˇ

ˇ D ˇ

ˇ

˚QS > QS .t˛/�ˇ

ˇ � t˛:

It follows easily that

kgtk2 � C

�Z 1

t˛QS .x/2 dx

�1=2

C C QS .t˛/t˛=2

and

Z 1

0

�

t1�� kgtk2�q dt

t� C

Z 1

0

t.1��/q�Z 1

t˛QS .x/2 dx

�q=2dt

t

C CZ 1

0

t.1��/q QS .t˛/qt˛q=2 dt

t

DW .A/C .B/:

First let us estimate .B/ by replacing u D t˛:

.B/ � CZ 1

0

u.1��/q=p0�.1��/q=2 QS .u/quq=2 du

uD C

S f

q

p;q:

In .A/, replace again u D t˛ and use (3.5.2):

.A/ � CZ 1

0

u.1��/q=p0�.1��/q=2�Z 1

u

QS .x/2 dx

�q=2du

u

� CZ 1

0

uq=p�q=2

�Z 1

u=4

�

x1=2 QS .x/�r dx

x

�q=rdu

u

� CZ 1

0

uq=p�q=2

�Z 1

uxr=2�1 QS .x/r dx

�q=rdu

u;


where r � min.2; q/. Applying Hardy’s inequality (1.7.3), we can conclude that

.A/ � CZ 1

0

�

uur=2�1 QS .u/r�q=r

uq=p�q=2 du

u

D CZ 1

0

QS .u/quq=p du

u

D C

S f

q

p;q:

Consequently,

k f k.Hp0 ;L2/�;qD�Z 1

0

t��K.t; f ;Hp0 ;L2/�q dt

t

�1=q

� C k f kHp;q:

If q D 1, then

supt>0

t�� khtkHp0� C sup

t>0t��

Z t˛

0

QS .x/p0 dx

!1=p0

D C supt>0

t��=˛�Z t

0

QS .x/p0xp0=px�p0=p dx

�1=p0

� C

S

p;1 supt>0

t��=˛�Z t

0

x�p0=p dx

�1=p0

D C

S

p;1 :

Moreover,

supt>0

t1�� kgtk2

� C supt>0

t1��Z 1

t˛QS .x/2 dx

�1=2

C C supt>0

t1��C˛=2 QS .t˛/

D C supt>0

t.1��/=˛�Z 1

t

QS .x/2 dx

�1=2

C C supt>0

t.1��/=˛C1=2 QS .t/

� C supt>0

t.1��/=˛�Z 1

t

QS .x/2x2=px�2=p dx

�1=2

C C

S

p;1

� C

S

p;1

as above because p < 2. Then

k f k.Hp0 ;L2/�;1D sup

t>0t��K.t; f ;Hp0 ;L2/ � C k f kHp;1

:

For the converse observe that

T W L2.Rd/ �! L2.R

d/ and T W Hp0 .Rd/ �! Lp0 .R

d/

are bounded, where T W f 7! S f . Therefore,

T W �Hp0 ;L2�

�;q�! �

Lp0 ;L2�

�;qD Lp;q

is bounded as well. In other words,

k f kHp;qD

S f

p;q � C k f k.Hp0 ;L2/�;q;

which proves the desired result. �Applying the reiteration and Wolff’s theorem (Theorems 1.7.9 and 1.7.10)

presented in Sect. 1.7 we get the following result.

Corollary 3.5.8 Suppose that 0 < � < 1 and 0 < p0; p1; q0; q1; q � 1. If p0 ¤ p1,then

�

Hp0;q0 ;Hp1;q1

�

�;qD Hp;q;

1

pD 1 � �

p0C �

p1:

In a special case,

�

Hp0 ;Hp1

�

�;p D Hp;1

pD 1 � �

p0C �

p1

and, for 1 < p1 � 1,

�

H1;Lp1

�

�;p D Lp;1

pD 1 � �C �

p1:

This result can be found in Weisz [346] and Lin [229] with another proof.

Corollary 3.5.9 If a sublinear or linear operator V is bounded from Hp0 .Rd/ to

Lp0 .Rd/ (resp. to Hp0 .R

d/) and from Lp1 .Rd/ to Lp1 .R

d/ . p0 � 1 < p1 � 1/, thenit is also bounded from Hp;q.R

d/ to Lp;q.Rd/ (resp. to Hp;q.R

d/) for each p0 < p < p1and 0 < q � 1.

3.6 Bounded operators on Hardy spaces 183

3.6 Bounded Operators on Multi-Dimensional Hardy Spaces

In this section we formulate the results of Sect. 1.8 for both Hardy spaces H�p .R

d/

and Hp.Rd/. Again, we can omit the proofs for H�

p .Rd/ because they are similar to

those of Sect. 1.8.

3.6.1 Bounded Operators on H�p .Rd/

Theorem 3.6.1 For each t 2 RdC let Vt W L1.Rd/ ! L1.Rd/ be a bounded linear

operator and let

V� f WD supt2Rd

C

jVt f j:

Suppose that

Z

RdnrIjV�ajp0 d� � Cp0

for all cube . p0; q/-atoms a and for some fixed r 2 N and 0 < p0 � 1, where thecube I is the support of the atom. If V� is bounded from Lp1 .R

d/ to Lp1 .Rd/ for some

1 < p1 � q � 1, then

kV� f kp � Cp k f kH�p

. f 2 H�p .R

d/\ L1.Rd// (3.6.1)

for all p0 � p � p1. If

limk!1 fk D f in the H�

p -norm implies that limk!1 Vt fk D Vt f in S0.Rd/

for all t 2 RdC, then (3.6.1) holds for all f 2 H�

p .Rd/.

By interpolation we obtain

Corollary 3.6.2 For each t 2 RdC let Vt W L1.Rd/ ! L1.Rd/ be a bounded linear

operator. Suppose that

Z

RdnrIjV�ajp0 d� � Cp0

for all cube . p0; q/-atoms a and for some fixed r 2 N and 0 < p0 < 1, where thecube I is the support of the atom. If V� is bounded from Lp1 .R

1 < p1 � q � 1, then


� �.jV� f j > �/ � C k f k1 . f 2 L1.Rd//:

Theorem 3.6.3 Suppose that Vf D f � K for all bounded tempered distributions,where K 2 L1.Rd/. If 0 0

�p�.jV�aj > �;Rd n rI/ � Cp

for all cube . p; q/-atoms a and for some fixed r 2 N and 0 < p < 1, where thecube I is the support of the atom. If V� is bounded from Lp1 .R


1 < p1 � q � 1, then

kV� f kp;1 � Cp k f kH�p

. f 2 H�p .R

d/ \ L1.Rd//: (3.6.2)

If


p -norm implies that limk!1 Vt fk D Vt f in S0.Rd/

for all t > 0, then (3.6.2) holds for all f 2 H�p .R

d/.

3.6.2 Bounded Operators on Hp.Rd/

The corresponding theorems for Hp.Rd/ are much more complicated. First we

consider the two-dimensional case. Since the definition of the p-atom is verycomplex, to obtain a usable condition about the boundedness of the operator, wehave to introduce simpler atoms (see also p. 157). Note that the results of thissubsection will be used later in Sect. 6.4.

Definition 3.6.5 Let d D 2. A function a 2 L2.R2/ is a simple p-atom or a rectanglep-atom if

(i) supp a � R for a rectangle R � R2,

(ii) kak2 � jRj1=2�1=p,(iii)

R

Ra.x/xk

i dxi D 0 for i D 1; 2, k D 0; : : : ;M. p/ D b2=p�3=2c and for almostevery fixed xj, j D 1; 2, j ¤ i.


Note that Hp.R2/ cannot be decomposed into rectangle p-atoms, a counterexam-

ple can be found in Weisz [347]. However, for the boundedness of V�, it will beenough to check the operator on these atoms. Before stating one of the main resultsof this section, we recall Journé’s covering lemma in one of its forms. Opposed tothe one-dimensional case, an open subset of R2 cannot be decomposed into disjointmaximal dyadic rectangles; however, the following lemma due to Journé [192, 193]holds.

Lemma 3.6.6 Let d D 2 and F be an open and bounded subset of R2. Assume thatthe dyadic rectangle R D I J belongs to M2.F/. Let F.1/ WD fMs.1F/ > 1=2gand I.1/ be the maximal dyadic interval containing I such that I.1/ J � F.1/, i.e.I.1/ J 2 M1.F.1//. Then

�

0

@

[

R2M2.F/

I.1/ J

1

A � CjFj (3.6.3)

and

X

R2M2.F/

� jIjjI.1/j

��

jRj � CjFj; (3.6.4)

for every � > 0, where C depends only on �, not on F.There is also a symmetric form of this lemma for rectangles in M1.F/. To the

proof of this lemma we need the following results.

Lemma 3.6.7 Let d � 2 and F � Rd be an open and bounded set. For a dyadic

interval I, set

EI.F/ WD[

˚

S � Rd�1 W S is dyadic and I S � F

�

:

Then

jFj DX

I

jIj jEI.F/ n E2I.F/j :

Proof Since

F D[

I

I .EI.F/ n E2I.F//�

;

the lemma follows from the fact that the sets fI .EI.F/ n E2I.F//g are disjoint. �


Lemma 3.6.8 Let d � 2, � > 0 and F � Rd be an open and bounded set. Then

X

I

jIj1X

kD0

� jIjjIkj

��

jEI.F/ n EIkC1 .F/j � CjFj:

Proof Let EI WD EI.F/. Since EI E2I : : : EIkC1 , we have

X

I

jIj1X

kD02�k� jEI n EIkC1 j

DX

I

jIj1X

kD02�k�

jEI n E2Ij C : : :C jEIk n EIkC1 j�

DX

I

jIj1X

kD02�k�

X

I0WI�I0�Ik

jEI0 n E2I0 j

D1X

kD02�k�

X

I0

jI0jjEI0 n E2I0 jX

IWI�I02�kjI0j�jIj

jIjjI0j

D1X

kD02�k�

X

I0

jI0jjEI0 n E2I0 j�

kX

jD0

X

IWI�I0jIjD2�j jI0j

2�j:

It follows from Lemma 3.6.7 that

X

I

jIj1X

kD02�k�jEI n EIkC1 j �

1X

kD0.k C 1/2�k�jFj � CjFj:

The proof of the lemma is complete. �Now we can prove Journé’s covering lemma.

Proof of Lemma 3.6.6 Inequality (3.6.3) follows easily:

jF.1/j � 4

Z

R2

Ms.1F/2 d� � C

Z

R2

12F d� D CjFj: (3.6.5)

For all k 2 N set

AI;kC1 WD[

˚

J W I J 2 M2.F/; I.1/ D Ik

�

:

Then

X

R2M2.F/

� jIjjI.1/j

��

jRj �X

I

jIj1X

kD0

X

JWIJ2M2.F/I.1/DIk

jJj� jIj

jI.1/j��

�X

I

jIj1X

kD02�k�

X

J2AI;kC1

jJj:

If J1; J2 � AI;kC1, then J1 and J2 are disjoint. Hence

X

R2M2.F/

� jIjjI.1/j

��

jRj �X

I

jIj1X

kD02�k�jAI;kC1j:

By Lemma 3.6.8, to prove (3.6.4) it is enough to show that

jAI;kC1j � C jEI.F/ n EIkC1 .F/j : (3.6.6)

Assume that x 2 AI;kC1. That is to say, x belongs to some dyadic J such that IJ � Fand I.1/ D Ik. Since Ik is the longest interval satisfying

jIk J \ Fj > 1

2jIk Jj;

we have

jIkC1 J \ IkC1 EIkC1 .F/j � jIkC1 J \ Fj � 1

2jIkC1 Jj:

This implies that

jJ \ EIkC1 .F/j � 1

2jJj:

Since J � EI.F/,

jJ \ .EI.F/ n EIkC1 .F//j > 1

2jJj

and so x 2 ˚Ms.1fEI.F/nEIkC1 .F/g/ > 1=2�

. Inequality (3.6.6) follows from (3.6.5). �The following result says that for an operator V� to be bounded from Hp.R

2/ toLp.R

2/ .0 < p � 1/ it is enough to check V� on simple p-atoms and the boundednessof V� on L2.R2/. Note that for the Hp.R

2/ spaces (as well as later for Hp.Rd/), we

have to suppose (3.6.7) for every r 2 P, while for the H�p .R

d/ spaces only for onefixed r (see Theorem 3.6.1).

Theorem 3.6.9 Let d D 2, 0 < p0 � 1 and for each t 2 R2C let Vt W L1.R2/ !

L1.R2/ be a bounded linear operator. Suppose that there exist � > 0 such that forevery simple p-atom a and for every r 2 P,

Z

.Rr/cjV�ajp0 d� � Cp02

��r; (3.6.7)

where R D I J is the support of a. If V� is bounded from L2.R2/ to L2.R2/, then

kV� f kp � Cpk f kHp . f 2 Hp.R2/ \ Hi

1.R2// (3.6.8)

for all p0 � p � 2 and i D 1; 2. If


k!1 Vt fk D Vt f in S0.R2/

for all t 2 R2C, then (3.6.8) holds for all f 2 Hp.R

2/.

Proof By the proof of Theorem 1.8.3, it is enough to prove that if a is a p0-atomthen

kV�akp0 � Cp0 : (3.6.9)

Let a be a p0-atom with support F. Set

F.1/ WD fMs.1F/ > 1=2g; F.2/ WD fMs.1F.1/ / > 1=2g

and

F.3/ WD fMs.1F.2// > 1=2g:

As in (3.6.5), we have

jF.3/j � CjF.2/j � CjF.1/j � CjFj:

Given a dyadic rectangle R D I J 2 M.F.1//, define the dyadic interval I.1/ suchthat

I.1/ I and R.1/ WD I.1/ J 2 M1.F.2//:

Furthermore, define the dyadic interval J.1/ such that

J.1/ J and R.2/ WD I.1/ J.1/ 2 M2.F.3//:


Set

2r1 WD 1.R;F.1// WD jI.1/j

jIj and 2r2 WD 2.R.1/;F.2// WD jJ.1/j

jJj :

Take the decomposition

a DX

R2M.F.1//

˛R

as in Definition 3.4.4. Then

Z

[R2M.F.1//R.2/

jV�ajp0 d� � �

0

@

[

R2M.F.1//

R.2/

1

A

1�p0=2�Z

R2

jV�aj2 d�

�p0=2

� �.F.3//1�p0=2�.F/p0=2�1 � Cp0 :

So we have to considerZ

R2n[R2M.F.1//

R.2/jV�ajp0 d� �

X

R2M.F.1//

Z

R2nR.2/jV�˛Rjp0 d�:

Obviously,

Z

R2nR.2/jV�˛Rjp0 d� �

Z

.RnI.1//R

jV�˛Rjp0 d�

CZ

R.RnJ.1//jV�˛Rjp0 d�:

Observe thatZ

.RnI.1//Œ0;1/jV�˛Rjp0 d� �

Z

R2n.I.1/Jr1 /

jV�˛Rjp0 d�

DZ

R2nRr1

jV�˛Rjp0 d�:

It is easy to see that the function

˛R

k˛Rk2jRj1=2�1=p0


is a rectangle p0-atom. By the condition of the theorem,

Z

.RnI.1//Œ0;1/jV�˛Rjp0 d� � Cp02

��r1 k˛Rkp02 jRj1�p0=2

D Cp01.R;F.1//�� k˛Rkp0

2 jRj1�p0=2:

Applying Hölder’s inequality and Journé’s lemma (see Lemma 3.6.6), we conclude

X

R2M.F.1//

Z

.RnI.1//Œ0;1/jV�˛Rjp0 d�

� Cp0

0

@

X

R2M.F.1//

k˛Rk22

1

A

p0=20

@

X

R2M.F.1//

1.R;F.1//�2�=.2�p0/jRj

1

A

1�p0=2

� Cp0 jFjp0=2�1jFj1�p0=2 D Cp0 :

Similarly,

X

R2M.F.1//

Z

R.RnJ.1//jV�˛Rjp0 d�

� Cp0

X

R2M.F.1//

2.R.1/;F.2//�� k˛Rkp0

2 jRj1�p0=2

� Cp0 jFjp0=2�10

@

X

R2M.F.1//

2.R.1/;F.2//�2�=.2�p0/jRj

1

A

1�p0=2

:

It is easy to see that if R1;R2 2 M.F.1// and R.1/1 D R.1/2 , then R1 \ R2 D ; orR1 D R2. Recall that R.1/ 2 M1.F.2//. So

X

R2M.F.1//

2.R.1/;F.2//�2�=.2�p0/jRj (3.6.10)

DX

S2M1.F.2//

0

@

X

R.1/DS

jRj1

A 2.S;F.2//�2�=.2�p0/

�X

S2M1.F.2//

jSj2.S;F.2//�2�=.2�p0/

� Cp0 jF.2/j � Cp0 jFj;


where we applied again Journé’s lemma and (3.6.5). Consequently,

X

R2M.F.1//

Z

R.RnJ.1//jV�˛Rjp0 d� � Cp0 ;

which proves (3.6.9) as well as the theorem. �Note that this result is due to Fefferman [105]. Unfortunately, the preceding proof

works for two dimensions, only. In the proof we decreased the dimension by 1 andwe used the fact that every one-dimensional open set can be decomposed into thedisjoint union of maximal dyadic intervals, which is obviously not true for higherdimensions. Journé [194] even verified that the preceding result do not hold fordimensions greater than 2. So there are fundamental differences between the theoryin the two-dimensional and three-dimensional or multi-dimensional cases. In thesequel we will extend the preceding theorem to higher dimensions as well (see alsoWeisz [355]).

First we consider the extension of Journé’s covering lemma to higher dimensions.We will state and prove this result due to Pipher [268] for d D 3, only, but it isreadily seen to extend to d > 3, inductively.

Lemma 3.6.10 Let d D 3 and F be an open and bounded subset of R3. Assumethat the dyadic rectangle R D I J Q belongs to M3.F/. Set

F.1/ WD fMs.1F/ > 1=2g and F.2/ WD fMs.1F.1// > 1=2g :

Let I.1/ be the maximal dyadic interval containing I such that I.1/ J Q 2M1.F.1//, and J.1/ be the maximal dyadic interval containing J such that I.1/ J.1/ Q 2 M2.F.2//. Then

�

0

@

[

R2M3.F/

I.1/ J.1/ Q

1

A � CjFj (3.6.11)

and

X

R2M3.F/

jRj� jIj

jI.1/j��1

� jJjjJ.1/j

��2

� CjFj (3.6.12)

for every �1; �2 > 0, where C depends only on �1; �2, but not on F.

Proof Similar to (3.6.5), inequality (3.6.11) follows from

jF.2/j � CjF.1/j � CjFj:


For all k 2 N set

AI;kC1 WD[˚

S W I S 2 M3.F/; I.1/ D Ik

�

:

Observe that if I S 2 M3.F/ and I.1/ D Ik, then S 2 M2.AI;kC1/. For S DJ K 2 M2.AI;kC1/, let J.0/ be the maximal dyadic interval containing J such thatQS WD J.0/ K 2 M1.A

.1/I;kC1/, where

A.1/I;kC1 WD ˚

Ms.1AI;kC1/ > 1=2

�

:

Consider the sum

X

R2M3.F/

� jIjjI.1/j

��1 � jJjjJ.0/j

��2

jRj

�X

I

jIj1X

kD0

X

SWIS2M3.F/I.1/DIk

jSj� jIj

jI.1/j��1

� jJjjJ.0/j

��2

�X

I

jIj1X

kD02�k�1

X

S2M2.AI;kC1/

jSj� jJj

jJ.0/j��2

:

By Lemma 3.6.6,

X

R2M3.F/

� jIjjI.1/j

��1� jJj

jJ.0/j��2

jRj � CX

I

jIj1X

kD02�k�1 jAI;kC1j:

Recall that J.1/ is the longest interval such that

ˇ

ˇI.1/ J.1/ Q \ F.1/ˇ

ˇ >1

2

ˇ

ˇI.1/ J.1/ Qˇ

ˇ :

Let us fix I.1/ D Ik. Since Ik AI;kC1 � F.1/, we have

ˇ

ˇI.1/ J.0/ Q \ F.1/ˇ

ˇ � ˇ

ˇIk J.0/ Q \ Ik AI;kC1ˇ

ˇ

D ˇ

ˇIkˇ

ˇ

ˇ

ˇJ.0/ Q \ AI;kC1ˇ

ˇ

>1

2

ˇ

ˇIk J.0/ Qˇ

ˇ ;

where in the last step we used the definition of J.0/. This means that J.0/ � J.1/.Hence

X

R2M3.F/

� jIjjI.1/j

��1� jJj

jJ.1/j��2

jRj �X

R2M3.F/

� jIjjI.1/j

��1� jJj

jJ.0/j��2

jRj

� CX

I

jIj1X

kD02�k�1 jAI;kC1j:

The inequality

jAI;kC1j � C jEI.F/ n EIkC1 .F/j :

can be proved in the same way as (3.6.6). We obtain as in the proof of Lemma 3.6.8that

X

I

jIj1X

kD02�k�1 jAI;kC1j � CjFj; (3.6.13)

which proves the lemma. �Now we are ready to extend the definition of the rectangle p-atoms to higher

dimensions.

Definition 3.6.11 Let d � 3. A function a 2 L2.Rd/ is called a simple p-atom ifthere exist Ii � R intervals, i D 1; : : : ; j for some 1 � j � d � 1, such that

(i) supp a � I1 : : : Ij A for some measurable set A � Rd�j,

(ii) kak2 � .jI1j � � � jIjjjAj/1=2�1=p,(iii)

R

Ra.x/xk

i dxi D R

A a d� D 0 for all i D 1; : : : ; j, k D 0; : : : ;M. p/ D b2=p �3=2c and almost every fixed x1; : : : ; xi�1; xiC1; : : : ; xd.

If j D d �1, we may suppose that A D Id is also an interval. Of course if a 2 L2.Rd/

satisfies these conditions for another subset of f1; : : : ; dg than f1; : : : ; jg, then it isalso called simple p-atom.

Notice that the condition in (3.6.7) can also be formulated as follows:

�Z

.Ir/cJCZ

.Ir/cJcCZ

I.Jr/cCZ

Ic.Jr/c

�

jV�ajp0 d� � Cp02��r:

For higher dimensions we generalize this form. As in the two-parameter case,Hp.R

d/ cannot be decomposed into simple p-atoms. The next theorem is due tothe author [351, 355].

Theorem 3.6.12 Let d � 3, 0 < p0 � 1 and for each t 2 RdC let Vt W L1.Rd/ !

L1.Rd/ be a bounded linear operator. Suppose that there exist �1; : : : ; �d > 0 such


that for every simple p-atom a and for every r1 : : : ; rd 2 P,

Z

.Ir11 /

c��.Irjj /

c

Z

AjV�ajp0 d� � Cp02

��1r1 � � � 2��jrj ;

where I1 : : : Ij A is the support of a. If j D d � 1 and A D Id is an interval,then we also assume that

Z

.Ir11 /

c��.Ird�1d�1 /

c

Z

.Id/cjV�ajp0 d� � Cp02

��1r1 � � � 2��d�1rd�1 :

If V� is bounded from L2.Rd/ to L2.Rd/, then

kV� f kp � Cpk f kHp . f 2 Hp.Rd/\ Hi

1.Rd// (3.6.14)

for all p0 � p � 2 and i D 1; : : : ; d. If


k!1 Vt fk D Vt f in S0.Rd/

for all t 2 RdC, then (3.6.14) holds for all f 2 Hp.R

d/.

Proof For simplicity, we prove the theorem only for three dimensions. It can beshown for higher dimensions inductively and with the same ideas. For an open setF � R

3 set

F.0/ WD F and F.i/ WD fMs.1F.i�1// > 1=2g ; i D 1; : : : ; 7:

It is clear that

ˇ

ˇF.7/ˇ

ˇ � Cˇ

ˇF.6/ˇ

ˇ � : : : � CjFj:

Given a dyadic rectangle R D I J K 2 M3.F.1//, define the dyadic intervalsI.1/, J.1/ and K.1/, I.2/ and J.2/, K.2/ such that

I.1/ I and R.1/ WD I.1/ J K 2 M1.F.2//;

J.1/ J and R.2/ WD I.1/ J.1/ K 2 M2.F.3//;

K.1/ K and R.3/ WD I.1/ J.1/ K.1/ 2 M3.F.4//;

I.2/ I.1/ and R.4/ WD I.2/ J.1/ K.1/ 2 M1.F.5//;

J.2/ J.1/ and R.5/ WD I.2/ J.2/ K.1/ 2 M2.F.6//;

K.2/ K.1/ and R.6/ WD I.2/ J.2/ K.2/ 2 M3.F.7//:


For all k 2 N set

AI;kC1 WD[

˚

S W I S 2 M3.F.1//; I.1/ D Ik

�

:

Recall that I S 2 M3.F.1// and I.1/ D Ik implies S 2 M2.AI;kC1/.By the proof of Theorem 1.8.3, to verify the theorem it is enough to show that if

a is a p0-atom, then

kV�akp0 � Cp0 :

Let a be a p0-atom with support F. Take the decomposition

a DX

R2M.F.1//

˛R

as in the definition of the atoms and set

F WD[

R2M.F/

R.6/:

Since V� is bounded on L2.R3/, Hölder’s inequality implies

Z

FjV�ajp0 d� � ˇ

ˇFˇ

ˇ

1�p0=2�Z

R3

jV�aj2 d�

�p0=2

� Cˇ

ˇFˇ

ˇ

1�p0=2 kakp02

� Cˇ

ˇF.7/ˇ

ˇ

1�p0=2 jFjp0=2�1 � Cp0 :

Thus we have to consider

Z

.F/cjV�ajp0 d� D

Z

.F/csup

t2R3C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Vt

0

@

X

R2M3.F.1//

˛R

1

A

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d� (3.6.15)

DZ

.F/csup

t2R3C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2M3.F.1//

Vt˛R1.R.6//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

DZ

.F/csup

t2R3C

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2M3.F.1//

Vt˛R

1.I.2//c C 1.J.2//c C 1.K.2//c

� 1.I.2//c1.J.2//c � 1.I.2//c1.K.2//c � 1.J.2//c1.K.2//c

C 1.I.2//c1.J.2//c1.K.2//c�

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�:


For the first term we have

Z

.F/csup

t2R3C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2M3.F.1//

Vt˛R1.I.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d� (3.6.16)

DZ

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

Q2M3.F.4//

X

R2M3.F.1//;R.3/DQ

Vt˛R1.I.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

DZ

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

Q2M3.F.4//

Vt

0

@

X

R2M3.F.1//;R.3/DQ

˛R

1

A 1.I.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

DZ

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

Q2M3.F.4//

Vt˛Q1.I.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

DZ

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

I.1/

1X

kD0

X

SWI.2/D.I.1//kI.1/S2M3.F.4//

Vt˛I.1/S1..I.1//k/c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�;

where

˛Q WDX

R2M3.F.1//;R.3/DQ

˛R .Q 2 M3.F.4///:

Notice that the support of ˛Q is contained in the dyadic rectangle 5Q. It is easy tosee that instead of the last term in (3.6.16) it is enough to investigate the expression

Z

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

I

1X

kD0

X

SWI.1/DIk

IS2M3.F.1//

Vt˛IS1.Ik/c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d� (3.6.17)

DZ

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

I

1X

kD0Vt

0

@

X

S2M2.AI;kC1/

˛IS

1

A 1.Ik/c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

�X

I

1X

kD0

Z

.Ik/c

Z

R2

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

V�

0

@

X

S2M2.AI;kC1/

˛IS

1

A

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�:


Again,

A.i/I;kC1 WD�

Ms.1A.i�1/I;kC1

/ > 1=2

�

.i D 1; 2/;

where A.0/I;kC1 D AI;kC1. For S D J K 2 M2.AI;kC1/, we define the dyadic intervalsJ.0/ and K.0/ such that

J.0/ J and QS WD J.0/ K 2 M1.A.1/I;kC1/;

K.0/ K and S.0/ WD J.0/ K.0/ 2 M2.A.2/I;kC1/:

Set

AI;kC1 WD[

S2M2.AI;kC1/

S.0/:

First we consider one part of the expression in (3.6.17), we integrate over AI;kC1in the second integral. Observe that if

b DX

S2M2.AI;kC1/

˛IS

then supp b � I AI;kC1 and

b

kbk2�jIjjAI;kC1j

�1=2�1=p0

is a simple p0-atom. Applying (3.4.2), (3.6.13) and (f) of Definition 3.4.4, we canconclude that

X

I

1X

kD0

Z

.Ik/c

Z

AI;kC1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

V�

0

@

X

S2M2.AI;kC1/

˛IS

1

A

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

� Cp0

X

I

1X

kD02��1k

�jIjjAI;kC1j�1�p0=2

X

S2M2.AI;kC1/

˛IS

p0

2

� Cp0

X

I

1X

kD02�2�1k=.2�p0/jIjjAI;kC1j

!1�p0=2


0

B

@

X

I

1X

kD0

X

S2M2.AI;kC1/

˛IS

2

2

1

C

A

p0=2

� Cp0 jFj1�p0=2jFjp0=2�1 � Cp0 :

For the remaining integral we have

X

I

1X

kD0

Z

.Ik/c

Z

.AI;kC1/c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

V�

0

@

X

S2M2.AI;kC1/

˛IS

1

A

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

�X

I

1X

kD0

X

S2M2.AI;kC1/

Z

.Ik/c

Z

.S.0//c

ˇ

ˇ

ˇV�˛IS

ˇ

ˇ

ˇ

p0d�:

Observe that

.S.0//c � .J.0//c R

[

R .K.0//c:


X

I

1X

kD0

X

S2M2.AI;kC1/

Z

.Ik/c

Z

.S.0//c

ˇ

ˇ

ˇV�˛IS

ˇ

ˇ

ˇ

p0d�

�X

I

1X

kD0

X

S2M2.AI;kC1/

Z

.Ik/c

Z

.J.0//c

Z

R

ˇ

ˇ

ˇV�˛IS

ˇ

ˇ

ˇ

p0d�

CX

I

1X

kD0

X

S2M2.AI;kC1/

Z

.Ik/c

Z

R

Z

.K.0//c

ˇ

ˇ

ˇV�˛IS

ˇ

ˇ

ˇ

p0d�:

Since

˛IS

k˛ISk2.jIjjSj/1=2�1=p0

is a simple p0-atom, the condition about V�, the two-dimensional version of Journé’scovering lemma (Lemma 3.6.6) and (3.6.13) imply

X

I

1X

kD0

X

S2M2.AI;kC1/

Z

.Ik/c

Z

.J.0//c

Z

R

ˇ

ˇ

ˇV�˛IS

ˇ

ˇ

ˇ

p0d�

� Cp0

X

I

1X

kD0

X

S2M2.AI;kC1/

2��1k

� jJjjJ.0/j

��2

.jIjjSj/1�p0=2 k˛ISkp02


� Cp0

0

@

X

I

1X

kD0

X

S2M2.AI;kC1/

2�2�1k=.2�p0/

� jJjjJ.0/j

�2�2=.2�p0/

jIjjSj1

A

1�p0=2

0

@

X

I

1X

kD0

X

S2M2.AI;kC1/

k˛ISk22

1

A

p0=2

� Cp0

X

I

1X


!1�p0=2

jFjp0=2�1

� Cp0 jFj1�p0=2jFjp0=2�1 � Cp0 :

Similarly,

X

I

1X

kD0

X

S2M2.AI;kC1/

Z

.Ik/c

Z

R

Z

.K.0//c

ˇ

ˇ

ˇV�˛IS

ˇ

ˇ

ˇ

p0d�

� Cp0

X

I

1X

kD0

X

S2M2.AI;kC1/

2��1k

� jKjjK.0/j

��3

.jIjjSj/1�p0=2 k˛ISkp02

� Cp0

0

@

X

I

1X

kD0

X

S2M2.AI;kC1/

2�2�1k=.2�p0/

� jKjjK.0/j

�2�3=.2�p0/

jIjjSj1

A

1�p0=2

0

@

X

I

1X

kD0

X

S2M2.AI;kC1/

k˛ISk221

A

p0

� Cp0 jFjp0=2�10

@

X

I

1X

kD0

X

S2M2.AI;kC1/

2�2�1k=.2�p0/

� jKjjK.0/j

�2�3=.2�p0/

jIjjSj1

A

1�p0=2

:

By Lemma 3.6.6 and the idea used in (3.6.10) we can see that

X

S2M2.AI;kC1/

� jKjjK.0/j

�2�3=.2�p0/

jSj

DX

Q2M1.A.1/

I;kC1/

0

@

X

QSDQ

jSj1

A

� jKjjK.0/j

�2�3=.2�p0/


DX

Q2M1.A.1/

I;kC1/

jQj� jKj

jK.0/j�2�3=.2�p0/

� Cp0 jA.1/I;kC1j� Cp0 jAI;kC1j:

Hence

X

I

1X

kD0

X

S2M2.AI;kC1/

Z

.Ik/c

Z

R

Z

.K.0//c

ˇ

ˇ

ˇV�˛IS

ˇ

ˇ

ˇ

p0d�

� Cp0

X

I

1X


!1�p0=2

jFjp0=2�1 � Cp0 ;

because of (3.6.13). Thus we have estimated the first term of (3.6.15):

Z

.F/csup

t2R3C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2M3.F.1//

Vt˛R1.I.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d� � Cp0 :

We consider also the fourth term of (3.6.15):

Z

.F/csup

t2R3C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2M3.F.1//

Vt˛R1.I.2//c1.J.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�:

Similarly as in (3.6.16), this term is equal to

Z

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

Q2M3.F.4//

Vt

0

@

X

R2M3.F.1//;R.3/DQ

˛R

1

A 1.I.2//c1.J.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

DZ

R3

supt2R3

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

Q2M3.F.4//

Vt˛Q1.I.2//c1.J.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

DZ

R3

supt2R3

C

ˇ

ˇ

ˇ

ˇ

ˇ

X

I.1/

1X

kD0

X

J.1/

1X

lD0

X

K.1/WI.2/D.I.1//k;J.2/D.J.1//lI.1/J.1/K.1/2M3.F.4//

Vt˛I.1/J.1/K.1/1..I.1//k/c1..I.1//l/c

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�:


Instead, it is enough to estimate the term

Z

R3

supn2N3

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

I

1X

kD0

X

J

1X

lD0

X

KWI.1/DIk;J.1/DJl

IJK2M3.F.1//

Vt˛IJK1.Ik/c1.Il/c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

�X

I

1X

kD0

X

J

1X

lD0

X

KWI.1/DIk;J.1/DJl

IJK2M3.F.1//

Z

.Ik/c

Z

.Jl/c

Z

R

ˇ

ˇ

ˇV�˛IJK

ˇ

ˇ

ˇ

p0d�;

which can be estimated further by

Cp0

X

I

1X

kD0

X

J

1X

lD0

X

KWI.1/DIk ;J.1/DJl

IJK2M3.F.1//

2��1k2��2l .jIjjJjjKj/1�p0=2 k˛IJKkp02

� Cp0

X

I

1X

kD0

X

J

1X

lD0

X

KWI.1/DIk ;J.1/DJl

IJK2M3.F.1//

2�2�1k=.2�p0/2�2�2l=.2�p0/

jIjjJjjKj!1�p0=2

0

B

B

B

@

X

I

1X

kD0

X

J

1X

lD0

X

KWI.1/DIk ;J.1/DJl

IJK2M3.F.1//

k˛IJKk22

1

C

C

C

A

p0=2

:

Therefore, by (3.6.12),

Z

.F/csup

t2R3C

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

X

R2M3.F.1//

Vt˛R1.I.2//c1.J.2//c

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p0

d�

� Cp0 jFj1�p0=2jFjp0=2�1 � Cp0 :

The other terms of (3.6.15) can be handled in the same way. This completes theproof of the theorem. �

Note that Theorem 3.6.3 is also valid for the Hp.Rd/ spaces. Using interpolation

we obtain

Corollary 3.6.13 Let d � 2. Besides the conditions of Theorems 3.6.9 or 3.6.12assume that 0 < p0 < 1. If f 2 Hi

1.Rd/ for some i D 1; : : : ; d, then


� �.jV� f j > �/ � C k f kHi1

. f 2 Hi1.R

d//:

Proof By Theorem 3.6.9 or 3.6.12, V� is bounded from Hp0 .Rd/ to Lp0 .R

d/ andfrom L2.R2/ to L2.R2/. Hence it is also bounded from Hp;q.R

d/ to Lp;q.Rd/ for each

p0 �/ � C k f kH1;1 � C k f kHi1;

where we have used Theorem 3.3.2. �

Chapter 4Multi-Dimensional Fourier Transforms

We study the theory of multi-dimensional Fourier transforms, namely, the inversionformula and the convergence of Fourier transforms. We formulate the analogousresults to those of Sects. 2.1–2.4 for higher dimensions. In the first section, weintroduce the Fourier transform for functions and for tempered distributions andgive the most important results. Since these proofs are very similar to those of theone-dimensional ones, we omit the proofs. In Sect. 4.3, we consider four types ofDirichlet integrals of multi-dimensional Fourier transforms, i.e. the cubic, triangular,circular and rectangular Dirichlet integrals. Using the analogous results for thepartial sums of multi-dimensional Fourier series proved in Sect. 4.2, we show thatthe Dirichlet integrals converge in the Lp.R

d/-norm to the function .1 < p < 1/.The multi-dimensional version of Carleson’s theorem is also verified.

4.1 Fourier Transforms

Definition 4.1.1 For f 2 S.Rd/, the Fourier transform and the inverse Fouriertransform of f is defined by

F f .�/ WDbf .�/ WD 1

.2�/d=2

Z

Rdf .t/ e�{t�� dt

�

� 2 Rd�

and

f _.�/ WDbf .��/ D 1

.2�/d=2

Z

Rdf .t/ e{t�� dt

�

� 2 Rd�

;

respectively.


203

204 4 Multi-Dimensional Fourier Transforms

Note that u � x and the Schwartz functions were defined in Sects. 3.1.1 and 3.2,respectively. The definition of the Fourier transform can be extended to functionsf 2 Lp.R

d/ .1 � p � 2/ and to tempered distributions f 2 S0.Rd/ exactly as inthe one-dimensional case in Chap. 2. All the results of Sects. 2.1 and 2.2 hold formulti-dimensional functions and tempered distributions, where the dilation is givenin the multi-dimensional case by

Ds f .t/ WD jsj�d=2 f .s�1t/ .t 2 Rd; s 2 R n f0g/:

We point out the next theorem, only.

Theorem 4.1.2 If f ; g; h 2 L2.Rd/, then

.a/Z

Rdf .x/bg.x/ dx D

Z

Rd

bf .x/g.x/ dx;

.b/

bf�_ D

2

�

f _�

D f ;

.c/Z

Rdf .x/h.x/ dx D

Z

Rd

bf .t/bh.t/ dt;

.d/ k f k2 D

bf

2D

f _

2:

4.2 Multi-Dimensional Partial Sums

The d-dimensional trigonometric system is introduced as a Kronecker product by

e{k�x DdY

jD1e{kjxj ;

where k D .k1; : : : ; kd/ 2 Zd, x D .x1; : : : ; xd/ 2 T

d.

Definition 4.2.1 For an integrable function f 2 L1.Td/ its kth Fourier coefficient isdefined by

bf .k/ D 1

.2�/d

Z

Tdf .x/e�{k�x dx .k 2 Z

d/:

The formal trigonometric series

X

k2Zd

bf .k/e{k�x .x 2 Td/

defines the multi-dimensional Fourier series of f .

4.2 Multi-Dimensional Partial Sums 205

We can generalize the partial sums in Definition 2.3.2 for multi-dimensionalfunctions basically in two ways. In the first version we replace the sum jkj � n inDefinition 2.3.2 by kkkq � n for some 1 � q � 1. In the literature the most naturalchoices q D 2 (see e.g. Stein and Weiss [309, 311], Davis and Chang [84], Grafakos[152, 154, 155], Lu and Yan [239], Feichtinger and Weisz [112, 113]), q D 1

(Berens, Li and Xu [26–28, 388], Weisz [363, 364]) and q D 1 (Marcinkiewicz[243], Zhizhiashvili [398] and Weisz [355, 365]) are investigated. In the secondgeneralization we take the sum in each dimension, the so-called rectangular partialsum (Zygmund [400] and Weisz [355]).

Definition 4.2.2 For f 2 L1.Td/ the nth `q-partial sum sqnf of the Fourier series of

f and the nth `q-Dirichlet kernel Dqn .n 2 N/ are given by

sqn f .x/ WD

X

k2Zd ;kkkq�n

bf .k/e{k�x

and

Dqn.u/ WD

X

k2Zd ; kkkq�n

e{k�u;

respectively.It is clear that

sqn f .x/ D 1

.2�/d

Z

Tdf .x � u/Dq

n.u/ du:

The partial sums are called triangular if q D 1, circular (or spherical) if q D 2 andcubic if q D 1 (see Figs. 4.1, 4.2, 4.3 and 4.4).

Fig. 4.1 Regions of the `q-partial sums for d D 2


−4−2

02

4

−4−2

02

4−10

0

10

20

30

40

50

Fig. 4.2 The Dirichlet kernel Dqn with d D 2, q D 1, n D 4

−4−2

02

4

−4−2

02

4−10

0

10

20

30

40

50



−4−2

02

4

−4−2

02

4−20

0

20

40

60

80

100


Definition 4.2.3 For f 2 L1.Td/ and n D .n1; : : : ; nd/ 2 Nd, the nth rectangular

partial sum sn f of the Fourier series of f and the nth rectangular Dirichlet kernel Dn

are introduced by

sn f .x/ WDX

jk1j�n1

� � �X

jkdj�nd

bf .k/e{k�x

and

Dn.u/ WDX

jk1j�n1

� � �X

jkdj�nd

e{k�u;

respectively.Again,

sn f .x/ D 1

.2�/d

Z

Tdf .x � u/Dn.u/ du

and

Dn.u/ D Dn1 .u1/ � � � Dnd.ud/;

where Dnj is the one-dimensional Dirichlet kernel (see Fig. 4.5).

−20

2

−2

0

2

0

20

40

60

Fig. 4.5 The rectangular Dirichlet kernel with d D 2, n1 D 3, n2 D 5

Definition 4.2.4 For some n D .n1; : : : ; nd/ 2 Nd, the function

n1X

k1D�n1

� � �ndX

kdD�nd

cke{k�x .x 2 Rd/

is said to be a trigonometric polynomial.By iterating the one-dimensional result, we get easily the next theorem.

Theorem 4.2.5 If f 2 Lp.Td/ for some 1 < p < 1, then

ksn f kp � Cp k f kp .n 2 Nd/

and

limn!1 sn f D f in the Lp.T

d/-norm:

Here n ! 1 means the Pringsheim convergence, i.e. min.n1; : : : ; nd/ ! 1.

Proof By one-dimensional analogue of this theorem (Theorem 2.3.3),

Z

T

jsn f .x/jp dx1

DZ

T

ˇ

ˇ

ˇ

ˇ

ˇ

Z

T

�Z

Td�1

f .t/.Dn2 .x2 � t2/ � � � Dnd.xd � td// dt2 � � � dtd

�

Dn1 .x1 � t1/ dt1

ˇ

ˇ

ˇ

ˇ

ˇ

p

dx1

� Cp

Z

T

ˇ

ˇ

ˇ

ˇ

Z

Td�1

f .t/.Dn2 .x2 � t2/ � � � Dnd.xd � td// dt2 � � � dtd

ˇ

ˇ

ˇ

ˇ

p

dt1:

Applying this inequality .d � 1/-times, we get the desired inequality of Theo-rem 4.2.5. The convergence is a consequence of this inequality and of the density oftrigonometric polynomials. �

A similar result holds for the triangular and cubic partial sums.

Theorem 4.2.6 If q D 1;1 and f 2 Lp.Td/ for some 1 < p < 1, then

sqn f

p� Cp k f kp .n 2 N/

and

limn!1 sq

n f D f in the Lp.Td/-norm:

If q D 2, then the same result is valid for p D 2.

Proof The result for q D 1 follows from Theorem 4.2.5. For q D 2, it is a basicresult of Fourier analysis. If q D 1, then we will prove the result for d D 2, only.The general case can be proved in the same way. Observe that

Z

T2

f .x; y/e�{kx�{ly dx dy D 2

Z

T2

f .u � v; u C v/e�{u.kCl/�{v.l�k/ dudv: (4.2.1)

If jkjCjlj � n on the left-hand side, then jkClj � n and jl�kj � n on the right-handside, hence

s1n f .x; y/ D 2s1n g.u; v/; (4.2.2)

where

g.u; v/ WD f .u � v; u C v/; x D u � v; y D u C v:

Thus

s1n f

pD 21C1=p

s1n g

p� Cp kgkp � Cp k f kp

shows the result for q D 1, too. �Since the characteristic function of the unit ball is not an Lp.R

d/ .1 < p <

1; p ¤ 2; d � 2/, multiplier (see Fefferman [102] or Grafakos [152, p. 743] or Luand Yan [239, p. 52]), we have

Theorem 4.2.7 If d � 2, q D 2 and 1 < p < 1, p ¤ 2, then the precedingtheorem is not true.

The analogue of Carleson’s theorem does not hold in higher dimensions for therectangular partial sums. However, it is true for the triangular and cubic partial sums(see Tevzadze [325] for p D 2, Fefferman [100, 101] and Grafakos [152, p. 231]).Let us denote by

sq� f WD supn2N

jsqn f j

the maximal operator.

Theorem 4.2.8 If q D 1;1 and f 2 Lp.Td/ for some 1 < p < 1, then

sq�f

p� Cp k f kp

and if 1 < p � 1, then

limn!1 sq

n f D f a.e.

Proof We will prove the theorem for d D 2 only. The proof for higher dimensionsis similar. Suppose first that q D 1 and

bf .k; l/ D 0 for l < k or k < 0: (4.2.3)

Let

fx. y/ WD f .x; y/ .x; y 2 T/

and observe by Fubini’s theorem that fx belongs to Lp.T/. Hence, by Theorem 2.3.8,

ks�fxkp � Cp k fxkp (4.2.4)

for almost every x 2 T. Set

hl.x/ WD bfx.l/ D 1

2�

Z

T

fx. y/e�{ly dy .l 2 Z/

and observe that

khlkp D Cp

�Z

T

ˇ

ˇ

ˇ

ˇ

Z

T

fx. y/e�{ly dy

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

� Cp

�Z

T

Z

T

jfx. y/jp dy dx

�1=p

D Cp k f kp :

Thus hl 2 Lp.T/. Since

bhl.k/ D 1

2�

Z

T

hl.x/e�{kx Dbf .k; l/;

it is clear by (4.2.3) that each hl .l 2 Z/ is a trigonometric polynomial. Moreprecisely,bhl.k/ vanishes if k < 0 or k > l. Consequently,

sn fx. y/ DX

jlj�n

hl.x/e{ly

DX

jlj�n

lX

kD0bf .k; l/e{kx

!

e{ly

DX

0�k�l�n

bf .k; l/e{kxC{ly

D s1n f .x; y/:

Hence (4.2.4) implies

ks1� f kp D�Z

T

Z

T

js�fx. y/jp dy dx

�1=p

� Cp

�Z

T

Z

T

j fx. y/jp dy dx

�1=p

D Cpkf kp;

which proves the theorem if (4.2.3) holds. Obviously, the same holds for functionsf for whichbf .k; l/ D 0 if l > k or l < 0 and we could also repeat the proof for theother quadrants.

Let us define the projections (see Fig. 4.6)

PC1 f .x; y/ WD

X

k2N

X

l2Zbf .k; l/e{kxC{ly;

PC2 f .x; y/ WD

X

k2Z

X

l2Nbf .k; l/e{kxC{ly;

Q1f .x; y/ WDX

l�jkjbf .k; l/e{kxC{ly

and

Qf .x; y/ WDX

l�k�0bf .k; l/e{kxC{ly:

Fig. 4.6 The projections PC

1 PC

2 , Q1 and Q

By (4.2.1) and Theorem 2.3.6, we conclude that

Q1 f .x; y/ D 2PC1 PC

2 g.u; v/

and

kQ1f kp D 21C1=p

PC1 PC

2 g

p� Cp

PC2 g

p� Cp kgkp � Cpkf kp;

where

g.u; v/ WD f .u � v; u C v/; x D u � v; y D u C v

and 1 < p < 1. Thus Q1 is a bounded projection on Lp.T2/ and so is Q D

Q1PC1 PC

2 . Since Qf satisfies (4.2.3), we obtain

ks1� .Q f /kp � Cp kQ f kp � Cp k f kp :

Each function f can be rewritten as the sum of eight similar projections, whichimplies the theorem for q D 1.

Equality (4.2.2) implies

s1� f

p� 21C1=p ks1� gkp � Cp kgkp � Cp k f kp ;

which also shows the result for q D 1. �Theorem 4.2.8 does not hold for circular partial sums (see Stein and Weiss [311,

p. 268] or Lu and Yan [239, p. 16]).

Theorem 4.2.9 If q D 2 and p < 2d=.dC1/, then there exists a function f 2 Lp.Td/

whose circular partial sums sqnf diverge almost everywhere.

In other words, for a general function in Lp.Td/ . p < 2/ almost everywhere

convergence of the circular partial sums is not true if the dimension is sufficientlylarge. It is an open problem, whether Theorem 4.2.8 holds for p D 2 and for circularpartial sums. As in the one-dimensional case, Theorem 4.2.5, Theorem 4.2.6 andthe inequality in Theorem 4.2.8 do not hold for p D 1 and p D 1.


4.3 Convergence of the Inverse Fourier Transform

The next result can be shown exactly in the same way as Corollary 2.4.1.

Corollary 4.3.1 If f 2 Lp.Rd/ for some 1 � p � 2 andbf 2 L1.Rd/, then for almost

every x 2 R,

f .x/ D 1

.2�/d=2

Z

R

bf .t/ e{t�x dt:

This motivates the definition of the Dirichlet integrals. As for Fourier series, theDirichlet integrals can be generalized basically in two ways for higher dimensions.In the first version we integrate over the set fktkq � Tg, T > 0, and in the secondversion over the rectangle Œ�T;T�, T 2 R

dC.

Definition 4.3.2 The Tth `q-Dirichlet integral sqT f of a function f 2 Lp.R

d/ .1 �p � 2/ and the Tth `q-Dirichlet kernel Dq

T .T > 0/ are given by

sqT f .x/ WD 1

.2�/d=2

Z

Rd1fktkq�Tgbf .t/e{x�t dt

and

DqT.x/ WD 1

.2�/d

Z

Rd1fktkq�Tge{x�t dt;

respectively.

Definition 4.3.3 The Tth rectangular Dirichlet integral sT f of the function f 2Lp.R

d/ .1 � p � 2/ and the Tth rectangular Dirichlet kernel DT .T 2 RdC/ are

given by

sT f .x/ WD 1

.2�/d=2

Z T1

�T1

� � �Z Td

�Td

bf .t/e{x�t dt

and

DT.x/ WD 1

.2�/d

Z T1

�T1

� � �Z Td

�Td

e{x�t dt;

respectively.The `q-Dirichlet integrals are called triangular if q D 1, circular if q D 2 and

cubic if q D 1 (see Figs. 4.7, 4.8, 4.9 and 4.10). It is easy to see that jDqT j � CTd

and jDT j � CTd . As in the one-dimensional case, sqT f and sT f are well defined

because of the Hausdorff-Young theorem.


−20

2

−2

0

2

0

0.5

1

Fig. 4.7 The Dirichlet kernel DqT with d D 2, q D 1, T D 4

−20

2

−2

0

2

0

0.2

0.4

0.6

0.8

1

1.2



−20

2

−2

0

2

0

0.5

1

1.5


−20

2

−2

0

2

0

0.5

1

Fig. 4.10 The rectangular Dirichlet kernel with d D 2, T1 D 3, T2 D 5

Proposition 4.3.4 We have DqT 2 Lp.R

d/ for q D 1;1 and DT 2 Lp.Rd/, whenever

1 2d=.d C 1/.

Proof Observe that

DT.x/ D DT1 .x1/ � � � DTd.xd/ DdY

jD1

sin Tjxj

�xj;

where DTj is the one-dimensional Dirichlet kernel. By the corresponding one-dimensional result DT 2 Lp.R

d/ .T 2 RdC/. Since D1

T D DT;:::;T .T > 0/,D1

T 2 Lp.Rd/. For q D 1 we show the result in the two-dimensional case, only.

Similar to (4.2.1),

Z

R2

1fktk1�Tge{.x1t1Cx2t2/ dt D 2

Z

R2

1fkuk1�T=2ge{u1.x1Cx2/C{u2.x2�x1/ du

and so

D1T.x/ D 2D1

T=2.x1 C x2; x2 � x1/: (4.3.1)

Thus

D1T

pD 21�1=p

D1T=2

p< 1: (4.3.2)

For q D 2 we will prove in the next section (see Corollary 4.4.11) that

D2T.x/ D .2�/�d=2Td=2kxk�d=2

2 Jd=2.Tkxk2/ .x 2 Rd/;

where jJk.t/j � Ckt�1=2 are the Bessel functions (k > �1=2, t > 0, seeLemma 4.4.8). Then

D2T.x/ � CTd=2�1=2kxk�d=2�1=2

2 :

The p-power of the right-hand side is integrable overRdnB.0; 1/ if p > 2d=.dC1/.�If f 2 L1.Rd/, then by Fubini’s theorem,

sqT f .x/ D 1

.2�/d

Z

Rd1fktkq�Tg

Z

Rdf . y/ e�{y�t dye{x�t dt

DZ

Rdf . y/Dq

T.x � y/ dy

and

sT f .x/ DZ

Rdf . y/DT.x � y/ dy:

Using Proposition 4.3.4, we can extend the definition of the Dirichlet integrals asin the one-dimensional case.

Definition 4.3.5 We extend the Tth `q-Dirichlet integral and the rectangular Dirich-let integral to the functions f 2 Lp.R

d/ .1 � p < 1/ by

sqT f .x/ WD

Z

Rdf . y/Dq

T.x � y/ dy .T > 0/

for q D 1;1, and by

sT f .x/ WDZ

Rdf . y/DT.x � y/ dy

�

T 2 RdC�

;

respectively. For q D 2 the definition holds for 1 � p < 2d=.d � 1/.The following two results can be shown as Theorems 4.2.5 and 4.2.6.

Theorem 4.3.6 If f 2 Lp.Rd/ for some 1 0/

and

limT!1 sq

T f D f in the Lp.Rd/-norm:

If q D 2, then the same result is valid for p D 2.

Theorem 4.3.8 If d � 2, q D 2 and 1 0

jsqT f j:

Carleson’s theorem for the triangular and cubic Dirichlet integrals is formulated asfollows. The proof follows from Theorem 4.2.8 and from the transference argumentpresented in the proof of Theorem 2.4.7. The analogous theorem for circular orrectangular Dirichlet kernels does not hold (cf. Theorem 4.2.9).

Theorem 4.3.9 If q D 1;1 and f 2 Lp.Rd/ for some 1 < p < 1, then

sq�f

p� Cp kf kp

and

limT!1 sq

T f D f a.e.

4.4 Multi-Dimensional Dirichlet Kernels

In this section we will give explicit forms for the Dirichlet kernels. These resultswill be used later several times. For q D 1 it is clear that

D1T .x/ D DT.x1/ � � � DT.xd/ D

dY

jD1

sin Txj

�xj.T > 0/:

The corresponding formula for DT .T 2 RdC/ is also true. The preceding formula

and (4.3.1) imply a formula for D1T ; however, we need a most finer description of

the Dirichlet kernels.

4.4.1 Triangular Dirichlet Kernels

To describe more precisely the Dirichlet kernels for q D 1, we need the concept ofdivided difference.

Definition 4.4.1 The nth divided difference of a one-dimensional function f at the(pairwise distinct) knots x1; : : : ; xn 2 R is introduced inductively as

Œx1� f WD f .x1/; Œx1; : : : ; xn� f WD Œx1; : : : ; xn�1� f � Œx2; : : : ; xn� f

x1 � xn: (4.4.1)

One can see that the difference is a symmetric function of the knots. If f is .n�1/-times continuously differentiable on Œa; b� and xi 2 Œa; b�, then by Rolle’s theoremthere exists � 2 Œa; b� such that

Œx1; : : : ; xn� f D f .n�1/.�/.n � 1/Š

: (4.4.2)

Moreover,

Œx1; : : : ; xn� f DnX

kD1

f .xk/Qn

jD1;j¤k.xk � xj/: (4.4.3)

4.4 Multi-Dimensional Dirichlet Kernels 219

For more about divided differences we refer DeVore and Lorentz [90, p. 120]. Thefollowing lemma was proved by Berens and Xu [27]. Let

GT.u/ WD .�1/Œd=2��du.d�2/=2soc .Tp

u/ .u > 0/;

where the function soc was defined in (2.10.2).

Lemma 4.4.2 We have

D1T.x/ D Œx21; : : : ; x

2d�GT D .�1/Œd=2��d

dX

kD1

xd�2k soc .xkT/

QdjD1;j¤k.x

2k � x2j /

: (4.4.4)

Proof First we note that the second equality follows from the definition of GT

and from (4.4.3). We prove the lemma by induction. It is known that in the one-dimensional case DT.x/ D sin.Tx/=.�x/, so (4.4.4) holds. Suppose the lemma istrue for an even d � 1. Then

D1T.x/

D ��dZ

.0;1/d1fkvk1�Tg cos.x1v1/ � � � cos.xdvd/ dv

D ��dZ T

0

Z T�v1

0

Z T�v1�v2

0

: : :

Z T�v1�:::�vd�1

0

cos.x1v1/ � � � cos.xdvd/ dv

D ��1Z T

0

cos.x1v1/D1T�v1 .x2; : : : ; xd/ dv1

D .�1/Œd=2��ddX

kD2

xd�3k

QdjD2;j¤k.x

2k � x2j /

Z T

0

cos.x1v1/ cos.xk.T � v1// dv1;

where D1T�v1 is the .d � 1/-dimensional kernel. Since

Z T

0

cos.x1v1/ cos.xk.T � v1// dv1

D 1

2

Z T

0

cos.x1v1 C xk.T � v1//C cos.x1v1 � xk.T � v1// dv1

D 1

2

sin.x1T/ � sin.xkT/

x1 � xkC 1

2

sin.x1T/C sin.xkT/

x1 C xk

D x1 sin.x1T/

x21 � x2k� xk sin.xkT/

x21 � x2k;


we have

D1T.x/ D .�1/Œd=2��d

dX

kD2

�xd�3k

QdjD1;j¤k.x

2k � x2j /

x1 sin.x1T/

CdX

kD2

xd�2k sin.xkT/

QdjD1;j¤k.x

2k � x2j /

!

:

Observe that by (4.4.2),

0 D Œx21; : : : ; x2d�id

.d�3/=2 DdX

kD1

xd�3k

QdjD1;j¤k.x

2k � x2j /

;

which shows the result (id is the identity function).If d � 1 is odd, then we apply

Z T

0

cos.x1v1/ sin.xk.T � v1// dv1

D 1

2

Z T

0

sin.x1v1 C xk.T � v1//� sin.x1v1 � xk.T � v1// dv1

D 1

2

cos.xkT/ � cos.x1T/

x1 � xk� 1

2

cos.xkT/ � cos.x1T/

x1 C xk

D xk cos.xkT/

x21 � x2k� xk cos.x1T/

x21 � x2k;

and the lemma can be proved in the same way. �We also need another representation of the kernel function D1

T . If we applythe inductive definition of the divided difference in (4.4.1) to D1

T , then in thedenominator we have to choose the factors from the following table with d � 1

rows:

x21 � x2dx21 � x2d�1 x22 � x2dx21 � x2d�2 x22 � x2d�1 x23 � x2d: : :

x21 � x2d�kC1 x22 � x2d�kC2 x23 � x2d�kC3 : : : x2k � x2d: : :

x21 � x22 x22 � x23 x23 � x24 : : : x2d�1 � x2d:

The last term in the kth row is x2k � x2d. Observe that the kth row contains k termsand the differences of the indices in the kth row are equal to d � k, more precisely,


if x2ik � x2jk is in the kth row, then jk � ik D d � k. We choose exactly one factor fromeach row. First we choose x21� x2d and then from the second row x21� x2d�1 or x22� x2d.If we have chosen the .k �1/th factor from the .k �1/th row, say x2j �x2jCd�kC1, thenwe have to choose the next one from the kth row as either the one below the .k�1/thfactor (it is equal to x2j �x2jCd�k) or its right neighbour (it is equal to x2jC1�x2jCd�kC1).

Definition 4.4.3 If the sequence of integer pairs ..in; jn/I n D 1; : : : ; d � 1/ hasthe following properties, then we say that it is in I. Let i1 D 1, j1 D d, .in/ isnon-decreasing and . jn/ is non-increasing. If .in; jn/ is given then let inC1 D in andjnC1 D jn � 1 or inC1 D in C 1 and jnC1 D jn.

Observe that the difference x2ik �x2jk is in the kth row of the table (k D 1; : : : ; d�1).So the factors we have just chosen can be written as

d�1Y

lD1.x2il � x2jl/:

In other words,

D1T.x/ D

X

.il;jl/2I.�1/id�1�1

d�2Y

lD1.x2il � x2jl/

�1Œx2id�1; x2jd�1

�GT ;

which proves

Lemma 4.4.4 We have

D1T.x/ D

X

.il;jl/2ID1

T;.il ;jl/.x/;

where

D1T;.il ;jl/.x/ D .�1/id�1�1

d�1Y

lD1.x2il � x2jl/

�1.GT.x2id�1/� GT.x

2jd�1//:

4.4.2 Circular Dirichlet Kernels

To give an explicit formula for D2T , we need the concept of Bessel functions. First,

we introduce the gamma function,

.x/ WDZ 1

0

tx�1e�t dt .x > 0/:


Integration by parts yields

.x/ D�

txe�t

x

�1

0

C 1

x

Z 1

0

txe�t dt D 1

x.x C 1/:

Since .1/ D 1, we have

.x C 1/ D x.x/ .x > 0/ and .n/ D .n � 1/Š: (4.4.5)


�

1

2

�

DZ 1

0

t�1=2e�t dt D 2

Z 1

0

e�u2 du D p�:

The beta function is defined by

B.x; y/ WDZ 1

0

sx�1.1 � s/y�1 ds DZ 1

0

sy�1.1� s/x�1 ds:

The relationship between the beta and gamma function is given in the next lemma.

Lemma 4.4.5 We have

.x C y/B.x; y/ D .x/. y/ .x; y > 0/:

Proof Substituting s D u=.1C u/, we obtain

.x C y/B.x; y/ D .x C y/Z 1

0

sy�1.1 � s/x�1 ds

D .x C y/Z 1

0

uy�1�

1

1C u

�xCy

du

DZ 1

0

Z 1

0

uy�1�

1

1C u

�xCy

vxCy�1e�v dvdu:

The substitution v D t.1C u/ in the inner integral yields

.x C y/B.x; y/ DZ 1

0

Z 1

0

uy�1txCy�1e�t.1Cu/ dt du

DZ 1

0

txe�tZ 1

0

.ut/y�1e�tu du dt

DZ 1

0

tx�1e�t. y/ dt

D .x/. y/;

which shows the lemma. �

Definition 4.4.6 For k > �1=2, the Bessel functions are defined by

Jk.t/ WD .t=2/k

.k C 1=2/.1=2/

Z 1

�1e{ts.1 � s2/k�1=2 ds .t 2 R/:

Note that the Bessel functions are real valued. We prove some basic properties ofthe Bessel functions.

Lemma 4.4.7 We have

J0k.t/ D kt�1Jk.t/ � JkC1.t/ .t ¤ 0/:

Proof By integrating by parts,

d

dt.t�kJk.t// D {2�k

.k C 1=2/.1=2/

Z 1

�1e{tss.1 � s2/k�1=2 ds

D {2�k

.k C 1=2/.1=2/

Z 1

�1{t

2k C 1e{ts.1 � s2/kC1=2 ds

D �2�k�1t.k C 1=2/.k C 1=2/.1=2/

Z 1

�1e{ts.1 � s2/kC1=2 ds

D �t�kJkC1.t/:

In the last step, we used (4.4.5). The lemma follows immediately. �

Lemma 4.4.8 For k > �1=2 and t > 0,

Jk.t/ � Cktk and Jk.t/ � Ckt�1=2;

where C is independent of t.

Proof The first estimate trivially follows from the definition of Jk. The second onefollows from the first one if 0 < t � 1. Assume that t > 1 and integrate the complexvalued function e{tz.1 � z2/k�1=2 .z 2 C/ over the boundary of the rectangle whoselower side is Œ�1; 1� and whose height is R > 0. By Cauchy’s theorem,

0 D {

Z 0

Re{t.�1C{s/.s2 C 2{s/k�1=2 ds C

Z 1

�1e{ts.1 � s2/k�1=2 ds

C {

Z R

0

e{t.1C{s/.s2 � 2{s/k�1=2 ds C �.R/;

where �.R/ ! 0 as R ! 1. Hence

Z 1

�1e{ts.1 � s2/k�1=2 ds D {e�{t

Z 1

0

e�ts.s2 C 2{s/k�1=2 ds

� {e{tZ 1

0

e�ts.s2 � 2{s/k�1=2 ds

DW I1 C I2:

Observe that

.s2 C 2{s/k�1=2 D .2{s/k�1=2 C �.s/;

where j�.s/j � CskC1=2 if 0 < s � 1 or s > 1 and k � 3=2. Moreover, j�.s/j �Cs2k�1 if s > 1 and k > 3=2. Indeed, by Lagrange’s mean value theorem

j�.s/j D ˇ

ˇ.2{s/k�1=2ˇˇ

ˇ

ˇ

ˇ

ˇ

s

2{C 1

�k�1=2 � 1ˇ

ˇ

ˇ

ˇ

� CkskC1=2ˇ

ˇ

ˇ

ˇ

�

2{C 1

ˇ

ˇ

ˇ

ˇ

k�3=2;

where 0 < � < s. Thus

js2 C 2{sjk�1=2 � Cksk�1=2 C j�.s/j

and

jI1j �Z 1

0

e�ts.Cksk�1=2 C j�.s/j/ ds

D Ckt�1Z 1

0

e�u.u=t/k�1=2 du CZ 1

0

e�tsj�.s/j ds CZ 1

1

e�tsj�.s/j ds:

The first term is Ck.k C 1=2/t�k�1=2, the second term can be estimated by

.k C 3=2/t�k�3=2 � Ckt�k�1=2

and the third one can be estimated by .k C 3=2/t�k�3=2 if k � 3=2 or by Cke�t ifk > 3=2, both are less than Ckt�k�1=2. The integral I2 can be estimated in the sameway. �

Lemma 4.4.9 If k > �1=2, l > �1 and t > 0, then

JkClC1.t/ D tlC1

2l.l C 1/

Z 1

0

Jk.ts/skC1.1 � s2/l ds:


Proof By Lemma 4.4.5,

Jk.t/ D 2.t=2/k

.k C 1=2/.1=2/

Z 1

0

cos.ts/.1 � s2/k�1=2 ds

D1X

jD0.�1/j 2.t=2/kt2j

.2j/Š.k C 1=2/.1=2/

Z 1

0

s2j.1 � s2/k�1=2 ds

D1X

jD0.�1/j .t=2/kt2j

.2j/Š.k C 1=2/.1=2/

Z 1

0

uj�1=2.1 � u/k�1=2 du

D1X

jD0.�1/j .t=2/kt2j

.2j/Š.k C 1=2/.1=2/B. j C 1=2; k C 1=2/

D .t=2/k

.1=2/

1X

jD0.�1/j . j C 1=2/

. j C k C 1/

t2j

.2j/Š:

Thus

Z 1

0

Jk.ts/skC1.1 � s2/l ds

DZ 1

0

0

@

.ts=2/k

.1=2/

1X

jD0.�1/j . j C 1=2/

. j C k C 1/

.ts/2j

.2j/Š

1

A skC1.1 � s2/l ds

D .t=2/k

.1=2/

1X

jD0.�1/j . j C 1=2/

. j C k C 1/

t2j

.2j/Š

Z 1

0

s2kC2jC1.1 � s2/l ds:

By a substitution and by Lemma 4.4.5, we conclude

Z 1

0

Jk.ts/skC1.1 � s2/l ds

D .t=2/k

.1=2/

1X

jD0.�1/j . j C 1=2/

2. j C k C 1/

t2j

.2j/Š

Z 1

0

ukCj.1 � u/l du

D .t=2/k

.1=2/

1X

jD0.�1/j . j C 1=2/

2. j C k C 1/

t2j

.2j/ŠB.k C j C 1; l C 1/

D 2l.l C 1/

tlC1.t=2/kClC1

.1=2/

1X

jD0.�1/j . j C 1=2/

.k C l C j C 2/

t2j

.2j/Š

D 2l.l C 1/

tlC1 JkClC1.t/;

which proves the lemma. �Let � be a one-dimensional even function and

�0.x/ WD �.kxk2/ .x 2 Rd/:

Then �0 is said to be a radial function. If �0 is radial, then its Fourier transform isalso radial and can be computed with the help of the Bessel functions.

Theorem 4.4.10 Suppose that

Z 1

0

j�.r/jrd�1 dr < 1:

Then for x 2 Rd and r D kxk2, we have

b�0.x/ D r�d=2C1Z 1

0

�.s/Jd=2�1.rs/sd=2 ds:

Proof Obviously, �0 2 L1.Rd/. Let r D kxk2, x D rx0, s D kuk2 and u D su0. Then

b�0.x/ D 1

.2�/d=2

Z

Rd�0.u/e

�{x�u du

D 1

.2�/d=2

Z 1

0

�.s/

�Z

†d�1

e�{rsx0�u0

du0�

sd�1 ds;

where †d�1 denotes the sphere. In the inner integral, we integrate first over theparallel Pı WD fu0 2 †d�1 W x0 � u0 D cos ıg orthogonal to x0 obtaining a function of0 � ı � � , which we then integrate over Œ0; ��. If !d�2 denotes the surface area of†d�2, then the measure of Pı is

!d�2.sin ı/d�2 D 2�.d�1/=2

..d � 1/=2/.sin ı/d�2:

HenceZ

†d�1

e�{rsx0�u0

du0 DZ �

0

e�{rs cos ı!d�2.sin ı/d�2 dı

D !d�2Z 1

�1e{rs�.1 � �2/.d�3/=2 d�

D 2�.d�1/=2

..d � 1/=2/.d=2� 1=2/.1=2/

.rs=2/d=2�1Jd=2�1.rs/

D .2�/d=2.rs/�d=2C1Jd=2�1.rs/;

which finishes the proof of the theorem. �


Using this theorem, the circular Dirichlet kernels can be written in a closed form.

Corollary 4.4.11 We have

D2T.x/ D .2�/�d=2Td=2kxk�d=2

2 Jd=2.Tkxk2/:

Proof Let us apply Theorem 4.4.10 for

� D 1.�T;T/ and �0.x/ D 1.�T;T/.kxk2/ D 1B.0;T/.x/:

By the definition of D2T ,

D2T.x/ D 1

.2�/d=211B.0;T/.x/

D .2�/�d=2kxk�d=2C12

Z T

0

Jd=2�1.kxk2s/sd=2 ds

D .2�/�d=2Td=2C1kxk�d=2C12

Z 1

0

Jd=2�1.Tkxk2s/sd=2 ds:

Using Lemma 4.4.9 with k D d=2� 1 and l D 0, we see that

D2T D .2�/�d=2Td=2kxk�d=2

2 Jd=2.Tkxk2/;

which shows the corollary. �

Chapter 5`q-Summability of Multi-Dimensional FourierTransforms

The summability of Fourier transforms can be generalized for higher dimensionsbasically in two ways. In this chapter, we study the `q-summability of higherdimensional Fourier transforms. As in the literature, we investigate the three casesq D 1, q D 2 and q D 1. The other type of summability, the so-calledrectangular summability, will be investigated in the next chapter. Both types aregeneral summability methods defined by a function � . We will generalize the resultsof Sects. 2.5–2.9. In the first section, we present the basic definitions of the `q-summability. In the next section, we prove the norm convergence of the �-means.It is shown that the maximal operator of the �-means is bounded from H�

p .Rd/

to Lp.Rd/ for any p > p0, which implies the almost everywhere convergence. In

Sect. 5.4, the convergence at Lebesgue points is investigated. Since the proofs arevery different for different q’s, therefore each case needs new ideas. Using the resultof the `1-summability, in the last section we prove the one-dimensional strongsummability results presented in Sect. 2.10.

5.1 The `q-Summability Means

As in the one-dimensional case, Theorems 4.3.7 and 4.3.9 are not true for p D 1

and p D 1. However, taking a summability method, we can extend the theorems top D 1 and p D 1 again.

Using a given one-dimensional function � , in this section we introduce thegeneral `q-�-summability method. Here we always assume that

� 2 C0.R/; �.k � kq/ 2 L1.Rd/; �.0/ D 1 and � is even: (5.1.1)

Suppose again first that f 2 Lp.Rd/ for some 1 � p � 2. Modifying slightly the

definition of the Dirichlet integral, we introduce the `q-�-means as follows.


229

230 5 `q-summability of Fourier transforms

Definition 5.1.1 The Tth `q-�-mean of the function f 2 Lp.Rd/ .1 � p � 2/ is

given by

�q;�T f .x/ WD 1

.2�/d=2

Z

Rd�

�ktkq

T

�

bf .t/e{x�t dt .x 2 Rd;T > 0/:

As in the one-dimensional case, the integral is well defined. The `q-�-meansare called triangular if q D 1, circular if q D 2 and cubic if q D 1. The cubicsummability (when q D 1) is also called Marcinkiewicz summability. Let us denoteby

�.q/0 .x/ WD �.kxkq/; �0 WD �

.2/0

�

x 2 Rd�

: (5.1.2)

For an integrable function f ,

�q;�T f .x/ D

Z

Rdf .x � t/Kq;�

T .t/ dt D f � Kq;�T .x/ .x 2 R

d;T > 0/:

Definition 5.1.2 The Tth `q-�-kernel is given by

Kq;�T .x/ WD 1

.2�/d

Z

Rd�

�ktkq

T

�

e{x�t dt D 1

.2�/d=2Tdb�

.q/0 .Tx/: (5.1.3)

For

�.t/ WD�

1 � jtj; if jtj � 1I0; if jtj > 1;

the kernel functions are called triangular-, circular- or cubic Fejér kernels (seeFigs. 5.1, 5.2 and 5.3).

It follows from Definition 5.1.2 thatˇ

ˇ

ˇKq;�T

ˇ

ˇ

ˇ � CTd .T > 0/ : (5.1.4)

Thus the `q-�-means can be rewritten as

�q;�T f .x/ D Td

.2�/d=2

Z

Rdf .x � t/

b

�.q/0 .Tt/ dt: (5.1.5)

For q D 2 we suppose in addition to (5.1.1) that

b

�.q/0 2 L1.R

d/: (5.1.6)

Since this condition is hard to check for other q’s, for q D 1 and q D 1, we startwith another concept and use the same conditions as in Sect. 2.10. Namely, suppose

5.1 The `q-summability means 231

−20

2

−2

0

2

0

0.2

0.4

0.6

0.8

1

1.2

Fig. 5.1 The Fejér kernel Kq;�T with d D 2, q D 1, T D 4

−20

2

−2

0

2

0

0.2

0.4

0.6

0.8

1


−20

2

−2

0

2

0

0.1

0.2

0.3

0.4


that � is even and absolutely continuous on R. Suppose further that

�.0/ D 1;

Z 1

0

.t _ 1/dj� 0.t/j dt < 1; limt!1 td�.t/ D 0 (5.1.7)

and assume thatˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/ ti .soc /.i/.tu/ dt

ˇ

ˇ

ˇ

ˇ

� Cu�˛ .i D 0; : : : ; d � 1/ (5.1.8)

for all u > 0 and for some 0 < ˛ < 1. Note that _ denotes the maximum and thefunction soc was defined earlier in (2.10.2). Since the integral on the left-hand sideis smaller than a constant C for every u, it is easy to see that if (5.1.8) holds for ˛,then it holds for all ˛0 � ˛. Observe that (5.1.7) implies that �.k � kq/ 2 L1.Rd/ forq D 1 and q D 1. Indeed, substituting

v1 C : : :C vd D x1; v2 D x2; : : : ; vd D xd;

we conclude

2�dZ

Rd�.kvk1/ dv D

Z

.0;1/d�.v1 C : : :C vd/ dv

DZ 1

0

Z x1

0

: : :

Z x1

0

�.x1/ dx

5.1 The `q-summability means 233

DZ 1

0

td�1�.t/ dt

D �1d

Z 1

0

td� 0.t/ dt:

Similarly,

1

C

Z

Rd�.kvk1/ dv D

Z

fv1>v2>:::>vd>0g�.v1/ dv

DZ 1

0

td�1�.t/ dt

D �1d

Z 1

0

td� 0.t/ dt:

By the first equality of (5.1.3),

Kq;�T .x/ D 1

.2�/d

Z

Rd�

�kukq

T

�

e{x�u du

D �1.2�/dT

Z

Rd

Z 1

kukq

� 0 t

T

�

dt e{x�u du

D �1.2�/dT

Z 1

0

� 0 t

T

�

Z

Rd1fkukq�tg e{x�u du dt

D �1T

Z 1

0

� 0 t

T

�

Dqt .x/ dt: (5.1.9)

Hence

�q;�T f .x/ D �1

T

Z 1

0

� 0 t

T

�

sqt f .x/ dt:

Note that for the Fejér means we get the definition

�q;�T f .x/ D 1

T

Z T

0

sqt f .x/ dt:

In the next section we will prove that

Z

Rd

ˇ

ˇ

ˇKq;�T .x/

ˇ

ˇ

ˇ dx � C .T > 0/:

Then we can extend the definition of the `q-�-means in the following way.


Definition 5.1.3 For q D 1; 2;1 we extend the Tth `q-�-mean to all f 2Lp.R

d/ .1 � p � 1/ or even to all f 2 W.L1; `1/.Rd/ by

�q;�T f WD f � Kq;�

T .T 2 RC/:

5.2 Norm Convergence of the `q-Summability Means

We show that the L1.Rd/-norms of the `q-�-kernels are uniformly bounded. Notethat in this and the next section we apply (5.1.8) for i D 0; 1, only.

Theorem 5.2.1 Under the conditions (5.1.1) and (5.1.6) with q D 2 or (5.1.7)and (5.1.8) with q D 1;1, we have

Z

Rd

ˇ

ˇ

ˇKq;�T .x/

ˇ

ˇ

ˇ dx � C .T > 0/:

Since b

�.2/0 2 L1.Rd/ by (5.1.6), the theorem is clear for q D 2. We will prove

this theorem in Sect. 5.2.1 for q D 1 and q D 1. Originally it was proved inBerens et al. [28], Oswald [264] and Weisz [363, 365, 370]. Moreover, Li andXu [218] investigated this theorem for Jacobi polynomials. The definition of thehomogeneous Banach spaces is extended to R

d is a usual way. Theorem 5.2.1implies easily.

Theorem 5.2.2 Assume that B is a homogeneous Banach space on Rd. Under the

conditions of Theorem 5.2.1,

�q;�T f

B� C k f kB .T > 0/

and

limn!1 �

q;�T f D f in the B-norm for all f 2 B:

Proof Since �.q/0 ;b

�.q/0 2 L1.Rd/ by Theorem 5.2.1, using (5.1.3) we conclude

�q;�T f .x/� f .x/ D 1

.2�/d=2

Z

Rd

f

x � t

T

�

� f .x/�

b

�.q/0 .t/ dt

and

�q;�T f � f

B� C

Z

Rd

T tT

f � f

B

ˇ

ˇ

ˇ

ˇ

b

�.q/0 .t/

ˇ

ˇ

ˇ

ˇ

dt:

5.2 Norm convergence of the `q-summability means 235

The theorem follows from the definition of the homogeneous Banach spaces andfrom the Lebesgue dominated convergence theorem. �

Since the spaces C0.Rd/, Lp.Rd/, the Lorentz spaces Lp;q.R

d/ .1 < p <

1; 1 � q < 1/, the Wiener amalgam spaces W.Lp; `q/.Rd/, W.Lp; c0/.Rd/,

W.C; `q/.Rd/ .1 � p; q < 1/, the Hardy spaces H�

1 .Rd/, H1.R

d/ and thespace Cu.R

d/ of uniformly continuous bounded functions endowed with thesupremum norm are all homogeneous Banach spaces, Theorem 5.2.2 holdsfor these spaces, too. Of course, the inequality of the theorem holds also forL1.Rd/.

5.2.1 Proof of Theorem 5.2.1 for q D 1 and q D 1

In this subsection we will prove Theorem 5.2.1. Since the kernel functions andhence the proofs are very different for different q’s, we prove the theorem infour subsections. Note that the idea described in (4.3.2) cannot be used for �-summability. For q D 1 and q D 1, we prove the theorem separately for d D 2 andd � 3.

5.2.1.1 Proof for q D 1 in the Two-Dimensional Case

Here we may write .x; y/ instead of the two-dimensional vector x. Lemma 4.4.2implies.

Lemma 5.2.3 We have

D1t .x; y/ D ��2 cos.yt/ � cos.xt/

.x � y/.x C y/:

In what follows, we may suppose that x > y > 0. We need the following sharpestimations of the kernel functions.

Lemma 5.2.4 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Then forall 0 � ˇ � 1,

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � C.x � y/�1�ˇ.x C y/�1Cˇ; (5.2.1)ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT�˛.x � y/�1�ˇy�˛Cˇ�1; (5.2.2)ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛y�1�˛: (5.2.3)

Proof Inequality (5.2.1) follows from Lemma 5.2.3 and from the fact that x C y >x � y. Indeed, by (5.1.9) we have

K1;�T .x; y/ D �

Z 1

0

� 0.t/D1Tt.x; y/ dt

D ��2Z 1

0

� 0.t/cos.yTt/ � cos.xTt/

.x � y/.x C y/dt

and soˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � C.x � y/�1�ˇ.x C y/�1Cˇ:

Since y < x, (5.1.8) implies

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT�˛.x � y/�1.x C y/�1y�˛ � CT�˛.x � y/�1�ˇy�˛Cˇ�1;

which is (5.2.2). By Lagrange’s theorem there exists y < u < x such that

cos.yTt/ � cos.xTt/ D Tt.x � y/ sin.Ttu/:

Thus

K1;�T .x; y/ D ��2T

Z 1

0

t� 0.t/sin.Ttu/

.x C y/dt

yields

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛.x C y/�1y�˛ � CT1�˛y�1�˛

and the proof is complete. �If 0 < ˛ � 1 in the preceding lemma, then

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � C.x � y/˛�1y�˛�1: (5.2.4)

Indeed, if T � .x � y/�1, then (5.2.2) with ˇ D 0 implies

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � C.x � y/˛�1y�˛�1:

If T < .x � y/�1, then we obtain the same inequality from (5.2.3).In the next lemma, we estimate the partial derivatives of the kernel function.

Lemma 5.2.5 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Then forall 0 � ˇ � 1 and j D 1; 2,

ˇ

ˇ

ˇ@jK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛.x � y/�1�ˇyˇ�1�˛:

Proof By Lemma 5.2.3 and Lagrange’s mean value theorem,

�2@1D1t .x; y/

D 4t sin.xt/.x � y/�1.x C y/�1

� 4.cos.yt/ � cos.xt//..x � y/�2.x C y/�1 C .x � y/�1.x C y/�2/

D 4t sin.xt/.x � y/�1.x C y/�1

� 4t sin.ut/..x � y/�1.x C y/�1 C .x C y/�2/;

where y < u < x is a suitable number. Using the methods above,

ˇ

ˇ

ˇ@1K1;�T .x; y/

ˇ

ˇ

ˇ Dˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/@1D1Tt.x; y/ dt

ˇ

ˇ

ˇ

ˇ

� CT1�˛.x � y/�1.x C y/�1y�˛ C CT1�˛.x C y/�2y�˛

� CT1�˛.x � y/�1�ˇyˇ�1�˛:

The inequality can be proved for j D 2 in the same way. �Now we are ready to prove that the L1.R2/-norm of the kernel functions is

uniformly bounded.

Proof of Theorem 5.2.1 for q D 1 and d D 2 By symmetry, it is enough to integratethe kernel function over the set f.x; y/ W 0 < y < xg. Let us decompose this set intothe union [5

iD1Ai, where

A1 WD f.x; y/ W 0 < x � 2=T; 0 < y < xg;A2 WD f.x; y/ W x > 2=T; 0 < y � 1=Tg;A3 WD f.x; y/ W x > 2=T; 1=T < y � x=2g;A4 WD f.x; y/ W x > 2=T; x=2 < y � x � 1=Tg;A5 WD f.x; y/ W x > 2=T; x � 1=T < y � xg:

The sets Ai can be seen on Fig. 5.4. Inequality (5.1.4) implies

Z

A1

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ dx dy � C:

Fig. 5.4 The sets Ai

By (5.2.1),

Z

A2

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ dx dy � CZ 1

2=T

Z 1=T

0

.x � 1=T/�1�ˇy�1Cˇ dy dx � C:

Since x � y > x=2 on the set A3 and ˇ can be chosen such that 0 < ˇ < ˛, we getfrom (5.2.2) that

Z

A3

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ dxdy � CT�˛Z 1

2=T

Z x=2

1=Tx�1�ˇy�˛Cˇ�1 dy dx � C:

Observe that y > x=2 on A4, hence (5.2.2) implies

Z

A4

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ


2=T

Z x�1=T

x=2.x � y/�1�ˇx�˛Cˇ�1 dy dx

� CT�˛CˇZ 1

2=Tx�˛Cˇ�1 dx

� C:

Finally, by (5.2.3),

Z

A5

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ dxdy � CT1�˛Z 1

1=T

Z yC1=T

yy�˛�1 dx dy

� CT�˛Z 1

1=Ty�˛�1 dy

� C;

which completes the proof of the theorem. �

5.2.1.2 Proof for q D 1 in Higher Dimensions (d � 3)

Lemma 4.4.4 and (5.1.9) prove

Lemma 5.2.6 We have

K1;�T .x/ D �

X

.il; jl/2I.�1/id�1�1

d�1Y

lD1.x2il � x2jl/

�1

Z 1

0

� 0.t/.GtT.x2id�1/� GtT.x

2jd�1// dt

DWX

.il; jl/2IK1;�

T;.il ; jl/.x/:

We may suppose that x1 > x2 > � � � > xd > 0. We will need the following sharpestimations of the kernel functions.

Lemma 5.2.7 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Then forall 0 < ˇ1; : : : ; ˇd�1 < 1 with

0 <

d�1X

lD1ˇl < ˛ C 1;

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ � CT�˛d�1Y

lD1.xil � xjl/

�1�ˇl xPd�1

lD1 ˇl�˛�1jd�1

:

Proof Since xil C xjl > xil � xjl and xil C xjl > xid�1 > xjd�1 , we can see that

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ

� Cd�1Y

lD1.x2il � x2jl/

�1ˇ

ˇ

ˇ

Z 1

0

� 0.t/.xd�2id�1

soc .tTxid�1 /� xd�2jd�1

soc .tTxjd�1 // dtˇ

ˇ

ˇ

� CT�˛d�1Y

lD1.x2il � x2jl/

�1.xd�2�˛id�1

C xd�2�˛jd�1

/

� CT�˛d�1Y

lD1.xil � xjl/

�1�ˇl .xd�2�˛CPd�1

lD1 .ˇl�1/id�1

C xd�2�˛CPd�1

lD1 .ˇl�1/jd�1

/

� CT�˛d�1Y

lD1.xil � xjl/

�1�ˇl xPd�1


;

which finishes the proof. �

Lemma 5.2.8 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Then forall 0 < ˇ1; : : : ; ˇd�2 < 1 with

0 <

d�2X

lD1ˇl < .˛ ^ 1/C 1;

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ � Cd�2Y

lD1.xil � xjl/

�1�ˇl

xPd�2

lD1 ˇl�1�˛jd�1

T1�˛ C xPd�2

lD1 ˇl�2jd�1

�

:

Proof Lagrange’s theorem and Lemma 4.4.4 imply that there exists x2id�1> � >

x2jd�1, such that

D1t;.il ; jl/

.x/ D .�1/id�1�1d�2Y

lD1.x2il � x2jl/

�1G0t.�/

D C.�1/id�1�1d�2Y

lD1.x2il � x2jl/

�1

�.d�3/=2soc 0.tp

�/t C �.d�4/=2soc .tp

�/�

:

Then

K1;�T;.il ; jl/

.x/ D Cd�2Y

lD1.x2il � x2jl/

�1

Z 1

0

� 0.t/

�.d�3/=2soc 0.tTp

�/tT C �.d�4/=2soc .tTp

�/�

dt:

We apply that jsoc j � 1 and (5.1.8) to obtain

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ � Cd�2Y

lD1.x2il � x2jl/

�1

�.d�3�˛/=2T1�˛ C �.d�4/=2�

:

Since xil C xjl > xil � xjl and xil C xjl > xid�1 > �1=2 > xjd�1 , we can see that

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ � Cd�2Y

lD1.xil � xjl/

�1�ˇl

�.d�3�˛CPd�2lD1 .ˇl�1//=2T1�˛ C �.d�4CPd�2

lD1 .ˇl�1//=2�

� Cd�2Y

lD1.xil � xjl/

�1�ˇl

xPd�2


T1�˛ C xPd�2

lD1 ˇl�2jd�1

�

:

�

Lemma 5.2.9 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ � 1. Then forall 0 < ˇ1; : : : ; ˇd�1 < 1 with

0 <

d�1X

lD1ˇl < ˛ C 1;

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ � Cd�1Y

lD1.xil � xjl/

�1�ˇl xPd�1


.xid�1 � xjd�1 /˛

C Cd�2Y

lD1.xil � xjl/

�1�ˇl xPd�2


.xid�1 � xjd�1 /˛�1

C Cd�2Y

lD1.xil � xjl/

�1�ˇl xPd�2

lD1 ˇl�2jd�1

DW K1;�.il; jl/;1

.x/C K1;�.il; jl/;2

.x/C K1;�.il; jl/;3

.x/:

Proof The result follows from

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ

�ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ 1T>.xid�1�xjd�1 /�1 C

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ 1T�.xid�1�xjd�1 /�1

and from Lemmas 5.2.7 and 5.2.8. �

In the next lemma, we estimate the partial derivatives of the kernel function.

Lemma 5.2.10 Suppose that 0 < ˇ1; : : : ; ˇd�1 < 1,

0 <

d�1X

lD1ˇl < ˛ C 1 and ˇl C ˇd�1

d � 2< 1

for all l D 1; : : : ; d � 2. If (5.1.7) and (5.1.8) hold for some 0 < ˛ � 1, then for alln D 1; : : : ; d,

ˇ

ˇ

ˇ@nK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ � Cd�1Y

lD1.xil � xjl/

�1�ˇl

xPd�1


T1�˛ C xPd�1

lD1 ˇl�2jd�1

�

:

Proof Let us write D1t;.il; jl/

.x/ in the form

D1t;.il; jl/.x/

D .�1/id�1�1d�1Y

lD1.xil � xjl/

�1.xil C xjl /�1.Gt.x

2id�1/� Gt.x

2jd�1//

(see Lemma 4.4.4). Computing the partial derivative of D1t;.il; jl/

, we have todifferentiate one of the factors

.xil � xjl/�1; .xil C xjl/

�1 and Gt.x2id�1/� Gt.x

2jd�1/

and sum up these derivatives. The coefficients are equal to 0 or ˙1. Let mi D0; 1 .i D 1; : : : ; d/, m0

i D 0; 1 .i D 1; : : : ; d � 1/ and ım1;:::;md ;m0

1;:::;m0

d�1D 0;˙1.

We differentiate the factor .xil � xjl/�1 ml-times, the factor .xil C xjl/

�1 m0l-times

.l D 1; : : : ; d � 1/ and the factor Gt.x2id�1/ � Gt.x2jd�1

/ md-times, depending on thefact, whether xn D xil or xn D xjl .l D 1; : : : ; d/. Then the partial derivative ofD1

t;.il; jl/can be written as

@nD1t;.il; jl/

.x/

D .�1/id�1�1 X

m1C:::CmdCm0

1C:::Cm0

d�1D1ım1;:::;md ;m0

1;:::;m0

d�1(5.2.5)

d�1Y

lD1.xil � xjl/

�1�ml .xil C xjl/�1�m0

l@mdn .Gt.x

2id�1/ � Gt.x

2jd�1//:

If we differentiate in the first .d�1/ factors, i.e. the factors .xil �xjl/�1, l D 1; : : : ; d�

1, then

X

m1C:::Cmd�1D1

d�1Y

lD1.xil � xjl/

�1�ml.xil C xjl/�1.Gt.x

2id�1/� Gt.x

2jd�1//

DX

m1C:::Cmd�1D1

d�1Y

lD1.xil � xjl/

�1�ml .xil C xjl/�1G0

t.�/.x2id�1

� x2jd�1/

and

X

m1C:::Cmd�1D1

d�1Y

lD1.xil � xjl/

�1�ml.xil C xjl/�1

ˇ

ˇ

ˇ

ˇ

Z 1

0


2jd�1// dt

ˇ

ˇ

ˇ

ˇ

� CX

m1C:::Cmd�1D1

d�2Y

lD1.xil � xjl/

�1.xil C xjl/�1.xid�1 � xjd�1 /

�1

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/



�/�

dt

ˇ

ˇ

ˇ

ˇ

:

This can be estimated further by

CX

m1C:::Cmd�1D1

d�2Y

lD1.xil � xjl /

�1�ˇl�ˇd�1=.d�2/.xil C xjl/�1CˇlCˇd�1=.d�2/

.xid�1 � xjd�1 /�1

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/



�/�

dt

ˇ

ˇ

ˇ

ˇ

� CX

m1C:::Cmd�1D1

d�2Y

lD1.xil � xjl/

�1�ˇl�ˇd�1=.d�2/.xid�1 � xjd�1 /�1

�.d�3�˛�.d�2/CPd�2lD1 ˇlCˇd�1/=2T1�˛ C �.d�4�.d�2/CPd�2

lD1 ˇlCˇd�1/=2�

� CX

m1C:::Cmd�1D1

d�1Y

lD1.xil � xjl/

�1�ˇl

xPd�1


T1�˛ C xPd�1

lD1 ˇl�2jd�1

�

;

whenever 0 <Pd�1

lD1 ˇl < ˛ C 1 and ˇl C ˇd�1

d�2 < 1.


For the derivative of the second .d � 1/ factors in (5.2.5), i.e. for the factors.xil C xjl/

�1 .l D 1; : : : ; d � 1/ we obtain

X

m0

1C:::Cm0

d�1D1

d�1Y

lD1.xil � xjl/

�1.xil C xjl/�1�m0

l

ˇ

ˇ

ˇ

ˇ

Z 1

0


2jd�1// dt

ˇ

ˇ

ˇ

ˇ

� CX

m0

1C:::Cm0

d�1D1

d�1Y

lD1.xil � xjl/

�1.xil C xjl/�1�m0

l

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/

xd�2id�1

soc .tTxid�1 /C xd�2jd�1

soc .tTxid�1 /�

dt

ˇ

ˇ

ˇ

ˇ

� CX

m0

1C:::Cm0

d�1D1

d�1Y

lD1.xil � xjl/

�1�ˇl xd�2CPd�1

lD1 ˇl�did�1

� CX

m0

1C:::Cm0

d�1D1

d�1Y

lD1.xil � xjl/

�1�ˇl xPd�1

lD1 ˇl�2jd�1

:

If we differentiate the factor Gt.x2id�1/ � Gt.x2jd�1

/ in (5.2.5), i.e. if md D 1 and,say id�1 D n, then

d�1Y

lD1.xil � xjl/

�1.xil C xjl/�[email protected]

2id�1/� Gt.x

2jd�1//

D Cd�1Y

lD1.xil � xjl/

�1.xil C xjl/�1

xd�2id�1

soc 0.txid�1 /t C xd�3id�1

soc .txid�1 /�

and

d�1Y

lD1.xil � xjl/

�1.xil C xjl/�1ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/@n.GtT.x2id�1/� GtT .x

2jd�1// dt

ˇ

ˇ

ˇ

ˇ

� Cd�1Y

lD1.xil � xjl/

�1�ˇ

xPd�1


T1�˛ C xPd�1

lD1 ˇl�2jd�1

�

:

Consequently,

ˇ

ˇ

ˇ@nK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ Dˇ

ˇ

ˇ

ˇ

�1T

Z 1

0

� 0 t

T

�

@nD1t;.il; jl/

.x/ dt

ˇ

ˇ

ˇ

ˇ

� Cd�1Y

lD1.xil � xjl/

�1�ˇxPd�1


T1�˛ C xPd�1

lD1 ˇl�2jd�1

�

;

which proves the lemma. �

Proof of Theorem 5.2.1 for q D 1 and d � 3 Since (5.1.8) holds for all ˛0 � ˛, weassume that ˛ � 1. By symmetry, we may suppose again that x1 > x2 > : : : > xd >

0. If x1 � 16=T, then (5.1.4) implies

Z

f16=T�x1>x2>:::>xd>0g

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ dx � C:

Let

S WD ˚

x 2 Rd W x1 > x2 > : : : > xd > 0; x1 > 16=T

�

:

For a sequence .il; jl/ 2 I let us define the set S.il; jl/;k by

S.il; jl/;k

WD�

x 2 S W xil � xjl > 4=T; l D 1; : : : ; k � 1; xik � xjk � 4=T; if k < dIx 2 S W xil � xjl > 4=T; l D 1; : : : ; d � 1; if k D d

and

S.il; jl/;k;1 WD�

x 2 S.il; jl/;k W xjk > 4=T; if k < dIx 2 S.il; jl/;k W xjd�1 > 4=T; if k D d;

S.il; jl/;k;2 WD�

x 2 S.il; jl/;k W xjk � 4=T; if k < dIx 2 S.il; jl/;k W xjd�1 � 4=T; if k D d:

ThenZ

Rd

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ 1S.x/dx

�dX

kD1

Z

Rd

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ .1S.il; jl/;k;1.x/C 1S.il; jl/;k;2.x// dx: (5.2.6)

We estimate the right-hand side in four steps.Step 1.

d�1X

kD1

Z

Rd

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ 1S.il; jl/;k;1 .x/ dx �3X

mD1

d�1X

kD1

Z

RdK1;�.il; jl/;m

.x/1S.il; jl/;k;1 .x/ dx:

Since xid�1 � xjd�1 � xil � xjl , Lemma 5.2.9 with ˇl D ˇ implyZ

RdK1;�.il; jl/;1

.x/1S.il; jl/;k;1 .x/ dx

� CZ

Rd

d�1Y

lD1.xil � xjl/

�1�ˇxˇ.d�1/�˛�1jd�1

.xid�1 � xjd�1 /˛1S.il; jl/;k;1.x/ dx

� CZ

Rd

k�1Y

lD1.xil � xjl/

�1�ˇ

d�1Y

lDk

.xil � xjl/�1�ˇC˛=.d�k/xˇ.d�1/�˛�1

jd�11S.il; jl/;k;1 .x/ dx:

First we choose the indices jd�1 .D i0d/, id�1 .D i0d�1/ and then id�2 if id�2 ¤ id�1 orjd�2 if jd�2 ¤ jd�1. (Exactly one case of these two cases is satisfied.) If we repeat thisprocess, then we get an injective sequence .i0l; l D 1; : : : ; d/. We integrate the termxi1�xj1 in xi01

, the term xi2�xj2 in xi02, . . . , and finally the term xid�1 �xjd�1 in xi0d�1

andxjd�1 in xi0d

. Since xil �xjl > 4=T .l D 1; : : : ; k�1/, xil �xjl � 4=T .l D k; : : : ; d�1/,xjd�1 � xjk > 4=T and we can choose ˇ such that ˇ < ˛=.d � 1/, we have

Z

RdK1;�.il; jl/;1


� Ck�1Y

lD1.1=T/�ˇ

d�1Y

lDk

.1=T/�ˇC˛=.d�k/.1=T/ˇ.d�1/�˛ � C: (5.2.7)

Similarly, if ˇ < ˛=.d � 1/, then

Z

RdK1;�.il; jl/;2

.x/1S.il; jl/;k;1.x/ dx

� CZ

Rd

k�1Y

lD1.xil � xjl/

�1�ˇd�2Y

lDk

.xil � xjl/�1�ˇC˛=.d�k/

.xid�1 � xjd�1 /˛=.d�k/�1xˇ.d�2/�˛�1

jd�11S.il; jl/;k;1 .x/ dx

� Ck�1Y

lD1.1=T/�ˇ

d�2Y

lDk

.1=T/�ˇC˛=.d�k/.1=T/˛=.d�k/.1=T/ˇ.d�2/�˛ � C;

and the same holds for the kernel K1;�.il; jl/;3

.x/, because it is equal to K1;�.il; jl/;2

.x/ with˛ D 1.

Step 2. For the dth summand we use Lemma 5.2.7 with ˇl D ˇ to obtain

Z

Rd

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ 1S.il; jl/;d;1 .x/ dx

� CT�˛Z

Rd

d�1Y

lD1.xil � xjl/

�1�ˇxˇ.d�1/�˛�1jd�1

1S.il; jl/;d;1.x/ dx

� CT�˛d�1Y

lD1.1=T/�ˇ.1=T/ˇ.d�1/�˛ � C;

if ˇ < ˛=.d � 1/.Step 3. Now let us consider the set S.il; jl/;k;2 in (5.2.6) .k D 1; : : : ; d � 1/. Then

xik � xjk � 4=T and so xjd�1 � xik � 8=T and this holds also for k D d. Observe thatk ¤ 1, because i1 D 1 and x1 > 16=T for the elements of S. Similar to (5.2.7), wecan conclude that

d�1X

kD2

Z

RdK1;�.il; jl/;1


� Ck�1Y

lD1.1=T/�ˇ

d�1Y

lDk

.1=T/�ˇC˛=.d�k/.1=T/ˇ.d�1/�˛ � C;

whenever ˇ.d � 1/� ˛ > 0 and ˇ < ˛=.d � k/ (k D 2; : : : ; d � 1).Moreover,

Z

RdK1;�.il; jl/;2


� CZ

Rd

d�2Y

lD1.xil � xjl/

�1�ˇxˇ.d�2/�˛�1jd�1

.xid�1 � xjd�1 /˛�11S.il; jl/;k;2 .x/ dx

� CZ

Rd

k�1Y

lD1.xil � xjl/

�1�ˇd�2Y

lDk

.xil � xjl/�1�ˇC.˛��/=.d�k�1/

.xid�1 � xjd�1 /��1xˇ.d�2/�˛�1

jd�11S.il; jl/;k;2 .x/ dx

� Ck�1Y

lD1.1=T/�ˇ

d�2Y

lDk

.1=T/�ˇC.˛��/=.d�k�1/.1=T/�.1=T/ˇ.d�2/�˛

� C;

whenever 0 < � < ˛ and ˛=.d � 2/ < ˇ < .˛ � �/=.d � 3/, which implies� < ˛=.d � 2/. The same holds again for the kernel K1;�

.il; jl/;3.x/.

Step 4. For the set S.il; jl/;d;2, we obtain similar to Step 2 that

Z

RdjK1;�

T;.il; jl/.x/j1S.il; jl/;d;2 .x/ dx

� CT�˛Z

Rd

d�1Y

lD1.xil � xjl/

�1�ˇxˇ.d�1/�˛�1jd�1

1S.il; jl/;d;2.x/ dx

� CT�˛d�1Y

lD1.1=T/�ˇ.1=T/ˇ.d�1/�˛

� C;

if ˇ > ˛=.d � 1/. The proof of the theorem is complete. �


We may suppose again that x > y > 0.

Lemma 5.2.11 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Then

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � Cx�1y�1; (5.2.8)ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT�˛x�1y�1.x � y/�˛; (5.2.9)ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛x�1.x � y/�˛: (5.2.10)

Proof We remind the validity of the trigonometric identities

sin a sin b D 1

2.cos.a � b/� cos.a C b//;

cos a sin b D 1

2.sin.a C b/� sin.a � b//:

(5.2.11)

Since

D1T .x; y/ D DT.x/DT.y/ D sin Tx

�x

sin Ty

�y.T > 0/;

by (5.1.9) we have

K1;�T .x; y/ D �

Z 1

0

� 0.t/cos.Tt.x � y//� cos.Tt.x C y//

2�2xydt:

This implies (5.2.8) and by (5.1.8),ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT�˛x�1y�1.x � y/�˛

because x � y � x C y. By Lagrange’s theorem there exists x � y < u < x C y suchthat

cos.Tt.x � y//� cos.Tt.x C y// D 2Tty sin.Ttu/:

Thus

K1;�T .x; y/ D �T

Z 1

0

t� 0.t/sin.Ttu/

�2xdt

yieldsˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛x�1.x � y/�˛;

which is exactly (5.2.10). �If T � y�1, then (5.2.9) implies

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � Cy˛�1.x � y/�˛�1: (5.2.12)

If in addition 0 < ˛ � 1, then T < y�1 and (5.2.10) imply the same inequality.

Lemma 5.2.12 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Thenfor j D 1; 2 and x > 1=T,

ˇ

ˇ

ˇ@jK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛x�1y�1.x � y/�˛:

Proof Since

�2@1D1t .x; y/ D y�1 sin.ty/

�

x�1t cos.tx/ � x�2 sin.tx/�

;

we obtain as above thatˇ

ˇ

ˇ@1K1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛x�1y�1.x � y/�˛ C CT�˛x�2y�1.x � y/�˛

� CT1�˛x�1y�1.x � y/�˛;

Proof of Theorem 5.2.1 for q D 1 and d D 2 We decompose the set f.x; y/ W 0 <y < xg again into the union of the sets

A1 WD f.x; y/ W 0 < x � 2=T; 0 < y < xg;A2 WD f.x; y/ W x > 2=T; 0 < y � 1=Tg;A3 WD f.x; y/ W x > 2=T; 1=T < y � x=2g;A4 WD f.x; y/ W x > 2=T; x=2 < y � x � 1=Tg;A5 WD f.x; y/ W x > 2=T; x � 1=T < y � xg

(see Fig. 5.4). It is clear that the integral of jK1;�T .x; y/j on A1 is uniformly bounded.

By (5.2.10),

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛.x � y/�1�˛

and so

Z

A2

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ dx dy � CT1�˛Z 1

2=T

Z 1=T

0

.x � 1=T/�1�˛ dy dx � C:

Since x � y > x=2 on the set A3, we get from (5.2.9) that

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT�˛x�1�˛y�1 � CT�˛x�1�˛Cˇy�1�ˇ (5.2.13)

for any 0 � ˇ. Thus

Z

A3

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ


2=T

Z x=2

1=Tx�1�˛Cˇy�1�ˇ dy dx � C

whenever 0 < ˇ < ˛. On the set A4, y > x=2 and y > x � y. Hence (5.2.2) implies

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ � CT�˛x�1y�1.x � y/�˛ � CT�˛x�1�ˇ.x � y/�1�˛Cˇ (5.2.14)

for any 0 � ˇ � 1. Then

Z

A4

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ


2=T

Z x�1=T

x=2.x � y/�1�˛Cˇx�1�ˇ dy dx

� CT�ˇZ 1

2=Tx�1�ˇ dx

� C

if 0 < ˇ < ˛. Finally, by (5.2.8),

Z

A5

ˇ

ˇ

ˇK1;�T .x; y/

ˇ

ˇ

ˇ dxdy � CZ 1

1=T

Z yC1=T

yx�2 dx dy � C;



We may suppose again that x1 > x2 > � � � > xd > 0. Recall that

D1t .x/ D

dY

jD1

sin txj

�xj:

Let � D .�1; : : : ; �d/ with �1 D 1 and �j D ˙1, j D 2; : : : ; d and �0 D.�1; : : : ; �d�1/. The sums

P

� andP

�0 meanP

�jD˙1;jD2;:::;d andP

�jD˙1;jD2;:::;d�1,respectively. Applying the trigonometric identities (5.2.11), we obtain

dY

iD1sin.xitT/ D 2�dC1X

�0

˙0

@soc

0

@tT

0

@

d�1X

jD1�jxj C xd

1

A

1

A

� soc

0

@tT

0

@

d�1X

jD1�jxj � xd

1

A

1

A

1

A :

Thus

K1;�T .x/ D �

Z 1

0

� 0.t/dY

iD1

sin.xitT/

�xidt

D �2�dC1��dX

�0

dY

iD1x�1

i

Z 1

0

� 0.t/

0

@soc

0

@tT

0

@

d�1X

jD1�jxj C xd

1

A

1

A

� soc

0

@tT

0

@

d�1X

jD1�jxj � xd

1

A

1

A

1

A dt

DWX

�0

K1;�T;�0 .x/: (5.2.15)

Lemma 5.2.13 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Then

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ � CdY

iD1x�1

i

and

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ � CX

�d

T�˛

dY

iD1x�1

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

:

Proof The lemma follows easily from (5.1.8). �Let us introduce the sets

S WD ˚

x 2 Rd W x1 > x2 > : : : > xd > 0; x1 > 32=T

�

;

Sk WD fx 2 S W x1 > x2 > : : : > xk � 4=T > xkC1 > : : : > xd > 0g

k D 1; : : : ; d and

S�0 WD8

<

:

x 2 Rd Wˇ

ˇ

ˇ

ˇ

ˇ

ˇ

d�1X

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< d � 16=T

9

=

;

;

S 0 WD8

<

:

x 2 Rd W 9�;

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< d � 16=T

9

=

;

;

S�;1 WD8

<

:

x 2 Rd Wˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< 4x1

9

=

;

;

S�0 ;d WD8

<

:

x 2 Rd Wˇ

ˇ

ˇ

ˇ

ˇ

ˇ

d�1X

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< 4xd

9

=

;

:

Recall that �1 D 1 and �j D ˙1, j D 2; : : : ; d.

Lemma 5.2.14 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ < 1. Thenfor all x 2 Sk n S�0 , x 2 Sk n S 0 .k D 1; : : : ; d � 1/ and x 2 Sc

�0 ;d,

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ � CX

�d

T1�˛

d�1Y

iD1x�1

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

:

Proof By Lagrange’s theorem

soc 0.tTu/2tTxd

D soc

0

@tT

0

@

d�1X

jD1�jxj C xd

1

A

1

A � soc

0

@tT

0

@

d�1X

jD1�jxj � xd

1

A

1

A

for some u 2 .Pd�1jD1 �jxj � xd;

Pd�1jD1 �jxj C xd/. If x 2 Sc

�0 ;d, then

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

d�1X

jD1�jxj C �dxd

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� 3xd:

In casePd�1

jD1 �jxj C xd � 0, we havePd�1

jD1 �jxj � xd � xd and so juj > jPd�1jD1 �jxj �

xdj. IfPd�1

jD1 �jxj C xd < 0, thenPd�1

jD1 �jxj � xd < 0 and juj > jPd�1jD1 �jxj C xdj. In

both cases

juj�˛ �ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

d�1X

jD1�jxj C xd

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

Cˇ

ˇ

ˇ

ˇ

ˇ

ˇ

d�1X

jD1�jxj � xd

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

:

The lemma can be proved in the same way if x 2 Sk n S�0 or x 2 Sk n S 0 .k D1; : : : ; d � 1/. �

Lemma 5.2.15 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ � 1. Then forall x 2 Sk n S�0 , x 2 Sk n S 0 .k D 1; : : : ; d � 1/ or x 2 Sc

�0 ;d,

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ � CX

�d

d�1Y

iD1x�1

i

!

x˛�1d

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

:

Proof The result follows from Lemma 5.2.13 if T � x�1d and from Lemma 5.2.14 if

T < x�1d . �

The partial derivatives of the kernel function are estimated in the next lemma.

Lemma 5.2.16 Assume that (5.1.7) and (5.1.8) hold for some 0 < ˛ � 1. Then forall l D 1; : : : ; d and x 2 S,

ˇ

ˇ

ˇ@lK1;�T;�0 .x/

ˇ

ˇ

ˇ � CX

�d

T1�˛

dY

iD1x�1

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

C CX

�d

dY

iD1x�1

i

!

x�1d 1[d�1

kD1.Sk\S�0 /[.Sd\S�0 ;d/.x/:

Proof Since

@

@xl

soc .xltT/

xlD tTsoc 0.xltT/

xl� soc .xltT/

x2l;

we obtain by Lemmas 5.2.13 and 5.2.14 that

ˇ

ˇ

ˇ@lK1;�T;�0 .x/

ˇ

ˇ

ˇ � CX

�d

T1�˛

dY

iD1x�1

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

C CX

�d

T1�˛

d�1Y

iD1x�1

i

!

x�1l

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛ d�1X

kD11SknS�0 .x/C 1SdnS�0 ;d.x/

!

C CX

�d

dY

iD1x�1

i

!

x�1l

d�1X

kD11Sk\S�0 .x/C 1Sd\S�0 ;d.x/

!

:

Now xl > xd finishes the proof. �

Proof of Theorem 5.2.1 for q D 1 and d � 3 We may suppose again that ˛ � 1

and x1 > x2 > : : : > xd > 0. If x1 � 32=T, then (5.1.4) implies

Z

f32=T�x1>x2>:::>xd>0g

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ dx � C:

It is enough to estimate the integrals

Z

S

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ dx

�X

kD1;d

Z

Sk\S0

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ dx CX

�0

d�1X

kD2

Z

Sk\S�0

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ dx

CX

kD1;d

Z

SknS0

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ dx CX

�0

d�1X

kD2

Z

SknS�0

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ dx: (5.2.16)

Step 1. It is easy to see in the first sum that if x 2 S 0, i.e.

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< d � 16=T;

then x1 must be in an interval of length d � 32=T. Since xk � 4=T > xkC1 on Sk, wehave

Z

S1\S0

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ dx � CTdZ

S1\S0

dx � C:

If k D d, then Lemma 5.2.13 implies

Z

Sd\S0

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ dx � CZ

Sd\S0

dY

iD1x�1

i dx

� CZ

Sd\S0

dY

iD2x�1�1=.d�1/

i dx

� CT�1T:

Step 2. Let us investigate the second sum in (5.2.16). First, we multiply by1S�0;d.x/ in the integrand. If x 2 S�0 ;d, then x1 is in an interval of length 8xd. ByLemma 5.2.13,

Z

Sk\S�0\S�0 ;d

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ dx

� CZ

Sk\S�0 \S�0;d

dY

iD1x�1

i dx

� CZ

Sk

kY

iD2x�1�1=.k�1/

i

!

dY

iDkC1x�1

i

!

xd dx2 : : : dxd

� CZ

Sk

kY

iD2x�1�1=.k�1/

i

!

dY

iDkC1x�1C1=.d�k/

i

!

dx2 : : : dxd

� CT=T:

If x 62 S�0 ;d, then jPdjD1 �jxjj � 3xd. Since jPd

jD1 �jxjj < d � 20=T on S�0 ,Lemma 5.2.15 implies

Z

Sk\S�0 nS�0 ;d

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ dx

� CX

�d

Z

Sk\S�0 nS�0 ;d

d�1Y

iD1x�1

i

!

x˛�1�ıd

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛Cı

dx

� CX

�d

Z

Sk\S�0

kY

iD2x�1�1=.k�1/

i

!

dY

iDkC1x�1C.˛�ı/=.d�k/

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛Cı

dx

� CT˛�ı�1T1�˛Cı ;

whenever ˛ � 1 < ı < ˛. This proves that the second sum in (5.2.16) is finite.Step 3. To estimate the fourth sum let us use Lemma 5.2.15:

Z

SknS�0

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ dx

� CX

�d

Z

.SknS�0 /\S�;1CZ

.SknS�0 /nS�;1

!

d�1Y

iD1x�1

i

!

x˛�1d

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

dx;

k D 2; : : : ; d � 1. Since jPdjD1 �jxjj � d � 12=T on Sc

�0 , we have

X

�d

Z

.SknS�0 /\S�;1

d�1Y

iD1x�1

i

!

x˛�1d

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

dx

� CX

�d

Z

SknS�0

kY

iD2x�1C.ı�1/=.k�1/

i

!

dY

iDkC1x˛=.d�k/�1

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

dx

� CX

�d

�

1

T

�ı�1 �1

T

�˛ �1

T

�1�˛�ı

� C;

whenever 1 � ˛ < ı � 1 and ı D 1 if k D 1. Similarly,

X

�d

Z

.SknS�0 /nS�;1

d�1Y

iD1x�1

i

!

x˛�1d

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

dx

� CX

�d

Z

Sk

x�1�˛1

d�1Y

iD2x�1

i

!

x˛�1d dx

� CX

�d

Z

Sk

kY

iD1x�1�˛=k

i

!

dY

iDkC1x˛=.d�k/�1

i

!

dx � C:


Step 4. In the third sum of (5.2.16) the inequality

Z

S1nS0

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ dx � C

can be estimated as in Step 3 with ı D 1. If k D d, then instead of Lemma 5.2.15we use Lemma 5.2.13 to show that

Z

SdnS0

ˇ

ˇ

ˇK1;�T;�0 .x/

ˇ

ˇ

ˇ dx � C;


5.2.2 Some Summability Methods

We have seen in Sect. 2.11 that Examples 2.11.3–2.11.11 all satisfy the condi-tions (5.1.1), (5.1.7) and (5.1.8). Thus all the results above are valid for thoseexamples and for q D 1 and q D 1. Here we consider some examples for q D 2.

Example 5.2.17 (Weierstrass Summation) Let

�0.t/ D e�ktk22=2 or �0.t/ D e�ktk2 .t 2 Rd/

(see Fig. 5.5). In the first case b�0.x/ D e�kxk22=2 (see Proposition 2.1.2) and in thesecond one, b�0.x/ D cd=.1 C kxk22/.dC1/=2 for some cd 2 R (see Stein and Weiss[311, p. 6.]).

Example 5.2.18 (Picard and Bessel Summations) Let

�0.t/ D 1

.1C ktk22/.dC1/=2 .t 2 Rd/

(see Fig. 5.6). Here b�0.x/ D cde�kxk2 for some cd 2 Rd.

Fig. 5.5 Weierstrasssummability function�0.t/ D e�ktk22=2

Fig. 5.6 Picard-Besselsummability function withd D 2

Fig. 5.7 Riesz summabilityfunction with d D 2, ˛ D 1, D 2

Example 5.2.19 (Riesz Summation) Let

�0.t/ D�

.1 � ktk2 /˛; if ktk2 > 1I0; if ktk2 � 1

.t 2 Rd/

for some 0 < ˛ < 1; 2 P (see Fig. 5.7). The summation is called Bochner-Rieszsummation if D 2.

Theorem 5.2.20 For the Bochner-Riesz summation,

b�0.x/ D 2˛.˛ C 1/ kxk�d=2�˛2 Jd=2C˛.kxk2/ .x 2 R

d/;

where Jd=2C˛ denotes the Bessel function (see Definition 4.4.6).

Proof By Theorem 4.4.10,

b�0.x/ D kxk�d=2C12

Z 1

0

Jd=2�1.kxk2s/sd=2.1 � s2/˛ ds:

Applying Lemma 4.4.9 with k D d=2� 1, l D ˛, we see that

b�0.x/ D kxk�d=2C12 Jd=2C˛.kxk2/ kxk�˛�1

2 2˛.˛ C 1/;

which shows the theorem. �Lemma 4.4.8 implies


Corollary 5.2.21 For the Bochner-Riesz summation, we haveˇ

ˇ

ˇ

b�0.x/ˇ

ˇ

ˇ � C kxk�d=2�˛�1=22 .x ¤ 0/:

The same result holds for 2 P (see Lu [233, p. 132]).

Corollary 5.2.22 In all examples of this subsection b�0 2 L1.Rd/ if

˛ >d � 1

2:

This implies that the results of Sect. 5.2 are true for these summability methods.

5.2.3 Further Results for the Bochner-Riesz Means

The situation is more complicated and exhaustively investigated in the literature, butnot completely solved for the Bochner-Riesz means if ˛ � .d � 1/=2. It is clear bythe Banach-Steinhaus theorem that

limT!1 �

q;�T f D f in the Lp.R

d/-norm

for all f 2 Lp.Rd/ if and only if the operators �q;�

T are uniformly bounded fromLp.R

d/ to Lp.Rd/. However, the norm of the operator

�q;�T W Lp.R

d/ ! Lp.Rd/

is equal to the one of

�q;�1 W Lp.R

d/ ! Lp.Rd/:

Indeed,

supk f kp�1

�Z

Rd

ˇ

ˇ

ˇ

ˇ

Z

Rd�

�ktkq

T

�

bf .t/e{x�t dt

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

D supk f kp�1

�Z

Rd

ˇ

ˇ

ˇ

ˇ

Z

RdTd�

�ktkq�

bf .Tt/e{x�Tt dt

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

D supk f kp�1

�Z

Rd

ˇ

ˇ

ˇ

ˇ

Z

RdTd.1�1=p/�

�ktkq�

bf .Tt/e{x�t dt

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

D supkgkp�1

�Z

Rd

ˇ

ˇ

ˇ

ˇ

Z

Rd��ktkq

�

bg.t/e{x�t dt

ˇ

ˇ

ˇ

ˇ

p

dx

�1=p

:


For simplicity, here we denote the Bochner-Riesz means by �2;˛T and the

Bochner-Riesz kernels by K2;˛T . Some special cases of the Bochner-Riesz means

can be seen in Figs. 5.8 and 5.9.The following results are all proved in the books of Davis and Chang [84],

Grafakos [152, 154, 155], Lu and Yan [239] and Stein and Weiss [309, 311], so we

−20

2

−2

0

2

0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 5.8 The Bochner-Riesz kernel K2;˛T with d D 2, T D 4, ˛ D 1, D 2

−20

2

−2

0

2

0

0.2

0.4

0.6

0.8

1

Fig. 5.9 The Bochner-Riesz kernel K2;˛T with d D 2, T D 4, ˛ D 1=10, D 2


do not prove them here. Figures 5.10, 5.11, 5.12, 5.13, and 5.14 show the regionswhere the operators �2;˛T are bounded or unbounded on the Lp.R

d/ spaces. Theboundedness on Lorentz spaces was investigated in Kim [199] and the boundednessof the bilinear Bochner-Riesz multiplier in [31].

Theorem 5.2.23 If d � 2, 0 � ˛ � .d � 1/=2 and

p � 2d

d C 1C 2˛or p � 2d

d � 1 � 2˛;

then the Bochner-Riesz operators �2;˛T are not bounded on Lp.Rd/ (see Fig. 5.10).

Fig. 5.10 Unboundedness of�2;˛T

Fig. 5.11 Boundedness of�2;˛T when d � 3


Fig. 5.12 Boundedness of �2;˛T when d D 2

Fig. 5.13 Boundedness of �2;˛T when d � 3

Fig. 5.14 Open question of the boundedness of �2;˛T when d � 3

The following result about the boundedness of �2;˛T was proved by Stein andWeiss [311, p. 276].

Theorem 5.2.24 If d � 2, 0 < ˛ � .d � 1/=2 and

2.d � 1/d � 1C 2˛

< p <2.d � 1/

d � 1 � 2˛ ;

then the Bochner-Riesz operators �2;˛T are bounded on Lp.Rd/ (see Fig. 5.11).

Approximately at the same time, Carleson and Sjölin [53] solved completely theboundedness of the Bochner-Riesz operators for d D 2. They are bounded for p’swhich are excluded from Theorem 5.2.23 (other proofs were given by Fefferman[103], Hörmander [186] and Cordoba [73, 74]).

Theorem 5.2.25 If d D 2, 0 < ˛ � 1=2 and

4

3C 2˛< p <

4

1 � 2˛;


Fefferman [99] generalized this result to higher dimensions.

Theorem 5.2.26 Suppose that d � 3. If

d � 1

2.d C 1/� ˛ � d � 1

2and

2d

d C 1C 2˛< p <

2d

d � 1 � 2˛;


Combining Theorems 5.2.26 and 5.2.24 and using analytic interpolation (seee.g. Stein and Weiss [311, p. 276, p. 205] or Lu and Yan [239]), we obtain


0 < ˛ <d � 12.d C 1/

and2.d � 1/

d � 1C 4˛< p <

2.d � 1/d � 1 � 4˛

;


It is still an open question as to whether the operators �2;˛T are bounded orunbounded in the region of Fig. 5.14.

5.3 Almost Everywhere Convergence of the `q-SummabilityMeans

Here we will use the conditions from the preceding section, more exactly, forq D 1;1 we assume the conditions (5.1.7) and (5.1.8) and for q D 2 theconditions (5.1.1) and (5.1.6).

Definition 5.3.1 The maximal operator �q;�� is defined by

�q;�� f WD sup

T>0

ˇ

ˇ

ˇ�q;�T f

ˇ

ˇ

ˇ :

Since b�.q/0 2 L1.Rd/, Eq. (5.1.5) implies

�q;�� f

1 � C

b

�.q/0

1

k f k1 . f 2 L1.Rd//: (5.3.1)

Using Theorems 3.6.1 and 3.6.4, we will show that the maximal operator isbounded from H�

p .Rd/ to Lp.R

d/ or to weak Lp.Rd/. Some versions of this result

were proved by Stein et al. [312] and Lu [233] for q D 2, by Oswald [264]for q D 1 and by the author [355, 360, 363, 365, 369, 370, 372] for q D1; 2;1.

Theorem 5.3.2 Suppose that q D 1;1 and � satisfies the conditions (5.1.7)and (5.1.8). If

p0 WD d

d C ˛ ^ 1 0

��.�q;�� f > �/1=p0 � C k f kH�

p0: (5.3.3)

For q D 2, we use again an additional condition. Similar to (2.8.1), we assumethat b�0 is .N C 1/-times differentiable for some N 2 N and there exists d C N <

ˇ � d C N C 1 such that

ˇ

ˇ

ˇ@i11 : : : @

iddb�0.x/

ˇ

ˇ

ˇ � C kxk�ˇ2 .x ¤ 0/; (5.3.4)

whenever i1 C : : :C id D N or i1 C : : :C id D N C 1.

Theorem 5.3.3 Suppose that q D 2 and � satisfies the conditions (5.1.1), (5.1.6)and (5.3.4). If d=ˇ 0

��.�q;�� f > �/ˇ=d � C k f kH�

d=ˇ:

If ˇ D d C N C 1 in (5.3.4), then it is enough to suppose that

ˇ

ˇ

ˇ@i11 : : : @

iddb�0.x/

ˇ

ˇ

ˇ � C kxk�d�N�12 .x ¤ 0/ (5.3.6)

for i1 C : : : C id D N C 1. The proof of the next result is similar to that ofTheorem 5.3.3 and is left to the reader.

Theorem 5.3.4 Suppose that q D 2 and � satisfies the conditions (5.1.1), (5.1.6)and (5.3.6). Then (5.3.5) holds for all d=.d C N C 1/ < p < 1 and for f 2

H�d=.dCNC1/.Rd/,

�q;�� f

d=.dCNC1/;1 D sup�>0

��.�q;�� f > �/.dCNC1/=d � C k f kH�

d=.dCNC1/

:

These theorems will be proved in the next two subsections. Recall thatH�

p .Rd/ � Lp.R

d/ for 1 < p � 1 and so we get that

�q;�� f

p� Cp k f kp . f 2 Lp.R

d/; 1 < p � 1/:

In some special cases it is known that Theorems 5.3.2 and 5.3.3 cannot be improved(see Oswald [264], Stein et al. [312]).

Theorem 5.3.5 Considering q D 1 with the Fejér summation, the operator �q;��is not bounded from H�

p .Rd/ to Lp.R

d/ if p � d=.d C 1/. Considering q D 2 and

the Bochner-Riesz summation with ˛ > .d � 1/=2, the operator �q;�� is not boundedfrom H�

p .Rd/ to Lp.R

d/ if p � d=.d=2C ˛ C 1=2/.Note that for the Fejér summation ˛ D 1 and for the Bochner-Riesz summation

ˇ D d=2C ˛ C 1=2 (for the last statement see Sect. 5.3.3).Marcinkiewicz [243] verified for two-dimensional Fourier series that the cubic

(i.e. q D 1) Fejér means of a function f 2 L log L.T2/ converge almost everywhereto f as n ! 1. Later Zhizhiashvili [396, 398] extended this result to all f 2 L1.T2/and to Cesàro means, Oswald [264] to Bochner-Riesz means and the author [365]to higher dimensions. The same convergence result for q D 2 can be found in Steinand Weiss [311], Lu [233] and Weisz [360], for q D 1 in Berens et al. [28] andWeisz [363, 364]. The next result is a consequence of Theorems 5.3.2, 5.3.3 andinterpolation theory.

Corollary 5.3.6 Assume the same conditions as in Theorems 5.3.2 or 5.3.3. If f 2L1.Rd/, then

�q;�� f

1;1 D sup�>0

� �.�q;�� f > �/ � Ck f k1:

This implies the almost everywhere convergence of the �-means in the usual way(cf. Corollary 2.7.7).

Corollary 5.3.7 Under the same conditions as in Theorems 5.3.2 or 5.3.3,

limT!1 �

q;�T f D f a.e.

for all f 2 Lp.Rd/ with 1 � p < 1

5.3.1 Proof of Theorem 5.3.2

In this section we will prove Theorem 5.3.2 in four subsections.


Proof of Theorem 5.3.2 for q D 1 and d D 2 As we mentioned before, if (5.1.8)holds for a number ˛, then it is true for all ˛0 � ˛. So we may assume that ˛ � 1.We will show that

Z

R2

ˇ

ˇ�1;�� a.x; y/ˇ

ˇ

pdx dy

DZ

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dx dy � Cp (5.3.7)

for every cube p-atom a, where 2=.2C˛/ < p < 1 and I is the support of the atom.By Theorem 3.6.1 and (5.3.1), this will imply (5.3.2). Without loss of generality, wecan suppose that a is a cube p-atom with support I D I1 I2 and

Œ�2�K�2; 2�K�2� � Ij � Œ�2�K�1; 2�K�1� . j D 1; 2/

for some K 2 Z. By symmetry, we can assume that x � u > y � v > 0, and so,instead of (5.3.7), it is enough to show that

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1Ai.x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dx dy � Cp

for all i D 1; 2; 3; 4; 5, where

A1 WD f.x; y/ W 0 < x � 2�KC5; 0 < y < xg;A2 WD f.x; y/ W x > 2�KC5; 0 < y � 2�KC2g;A3 WD f.x; y/ W x > 2�KC5; 2�KC2 < y � x=2g;A4 WD f.x; y/ W x > 2�KC5; x=2 < y � x � 2�KC2g;A5 WD f.x; y/ W x > 2�KC5; x � 2�KC2 < y < xg:

These sets are similar to those in Theorem 5.2.1 (see Fig. 5.4). First of all, if 0 <x � u � 2�KC5, then �2�K�1 < x � 2�KC6 and the same holds for y. By the

definition of the atom and by Theorem 5.2.1,

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A1 .x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dx dy

�Z

R2

22K supT>0

ˇ

ˇ

ˇ

ˇ

Z

I

ˇ

ˇ

ˇK1;�T .x � u; y � v/

ˇ

ˇ

ˇ 1A1.x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dx dy

� Cp22K2�2K :

Considering the set A2, we have x � u > 2�KC5 and 0 < y � v � 2�KC2, thus

2�KC4 < x < 1 and � 2�K�1 < y � 2�KC3:

Using (5.2.1), we conclude

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A2.x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

� Cp22K=p

Z

I.x � u � y C v/�1�ˇ

.x � u C y � v/�1Cˇ1A2.x � u; y � v/ dudv

� Cp22K=p1f2�KC4<xg1f�2�K�1<y�2�KC3g

Z

I.x � 2�KC3/�1�ˇ.2�KC5 C y � v/�1Cˇ dudv (5.3.8)

� Cp22K=p�K1f2�KC4<xg1f�2�K�1<y�2�KC3g.x � 2�KC3/�1�ˇ2�Kˇ

and so

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A2.x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dx dy

� Cp22K�Kp�Kˇp

Z 1

2�KC4

Z 2�KC3

�2�K�1

.x � 2�KC3/�p.1Cˇ/ dydx

� Cp;

provided that p > 1=.1C ˇ/ D 2=.2C ˛/ and ˇ D ˛=2.Since x � u � y C v > .x � u/=2 on the set A3, we apply (5.2.2) to obtain

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A3 .x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

� CpT�˛22K=pZ

I.x � u/�1�ˇ.y � v/�˛Cˇ�11A3.x � u; y � v/ dudv

� CpT�˛22K=p�2K1f2�KC4<xg1f2�KC1<y�x=2C2�Kg

.x � 2�K�1/�1�ˇ.y � 2�K�1/�˛Cˇ�1

� Cp22K=p�2K�K˛1f2�KC4<xg1f2�KC1<y�x=2C2�Kg

.x � 2�K�1/�1�ˇ.y � 2�K�1/�˛Cˇ�1; (5.3.9)

whenever T > 2K .Assume that T � 2K . We may suppose that the centre of I is zero, in other words

I WD .��; �/ .��; �/ for some � > 0. Let

A1.u; v/ WDZ u

��a.t; v/ dt and A2.u; v/ WD

Z v

��A1.u; t/ dt:

Observe that

jAk.u/j � Cp2K.2=p�k/ .k D 1; 2/:

Integrating by parts, we can see that

Z

I1

a.u; v/K1;�T .x � u; y � v/1A3 .x � u; y � v/ du

D A1.�; v/K1;�T .x � �; y � v/1A3 .x � �; y � v/

CZ �

��A1.u; v/@1K

1;�T .x � u; y � v/1A3.x � u; y � v/ du;

because A1.��; v/ D 0. As A2.�; �/ D R

I a D 0, integrating the first term again byparts, we obtain

Z

I1

Z

I2

a.u; v/K1;�T .x � u; y � v/1A3 .x � u; y � v/ dudv

DZ �

��A2.�; v/@2K

1;�T .x � �; y � v/1A3 .x � �; y � v/ dv

CZ

I1

Z

I2

A1.u; v/@1K1;�T .x � u; y � v/1A3.x � u; y � v/ dudv:

Lemma 5.2.5 implies

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A3 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

� CpT1�˛22K=p�2KZ

I2

.x � �/�1�ˇ.y � v/�˛Cˇ�11A3.x � �; y � v/ dv

C CpT1�˛22K=p�KZ

I.x � u/�1�ˇ.y � v/�˛Cˇ�11A3.x � u; y � v/ dudv

� CpT1�˛22K=p�3K1f2�KC4<xg1f2�KC1<y�x=2C2�Kg

.x � 2�K�1/�1�ˇ.y � 2�K�1/�˛Cˇ�1


.x � 2�K�1/�1�ˇ.y � 2�K�1/�˛Cˇ�1: (5.3.10)

Hence

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A3 .x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dxdy

� Cp22K�2Kp�K˛p

Z 1

2�KC4

Z x=2C2�K

2�KC1

.x � 2�K�1/.�1�ˇ/p.y � 2�K�1/.�˛Cˇ�1/p dydx

� Cp22K�2Kp�K˛p2�K.1�.1Cˇ/p/2�K.1�.˛�ˇC1/p/

� Cp;

whenever p > 1=.1Cˇ/ and p > 1=.˛�ˇC 1/, in other words, with ˇ D ˛=2 wehave

p >2

2C ˛:

Since y � v > .x � u/=2 on A4, (5.2.2) implies

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A4 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

� CpT�˛22K=p

Z

I.x � u � y C v/�1�ˇ.x � u/�˛Cˇ�11A4.x � u; y � v/ dudv

� CpT�˛22K=p�2K1f2�KC4<xg1fx=2�2�K<y�x�2�KC1g

.x � y � 2�K/�1�ˇ.x � 2�K�1/�˛Cˇ�1; (5.3.11)

thusZ

R2

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A4 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

p

dxdy


Z 1

2�KC4

Z x�2�KC1

x=2�2�K.x � y � 2�K/.�1�ˇ/p.x � 2�K�1/.�˛Cˇ�1/p dydx

� Cp22K�2Kp�K˛p2�K.1�.1Cˇ/p/2�K.1�.˛�ˇC1/p/

� Cp;

whenever ˇ D ˛=2 and p > 22C˛ .

If T � 2K and .x; y/ 2 A4, then we can prove the inequalityZ

R2

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A4.x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

p

dxdy � Cp

in the same way as before.Finally, if .x � u; y � v/ 2 A5, then x � u � y C v < 2�KC2. Inequality (5.2.4)

impliesˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A5 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

� Cp22K=p

Z

I.x � u � y C v/˛�1.y � v/�˛�11A5.x � u; y � v/ dudv

� Cp22K=p�K˛1f2�KC4<xg1fx�2�KC3<y�xC2�Kg

Z

I2

.y � 2�K�1/�˛�1 dv

� Cp22K=p�K�K˛1f2�KC4<xg1fx�2�KC3<y�xC2�Kg.y � 2�K�1/�˛�1;

whence we get that

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A5 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

p

dxdy

� Cp22K�Kp�K˛p

Z 1

2�KC4

Z xC2�K

x�2�KC3

.y � 2�K�1/�.˛C1/p dydx


Z 1

2�KC3

Z yC2�KC3

y�2�K.y � 2�K�1/�.˛C1/p dxdy

� Cp;

whenever p > 1=.˛ C 1/, which finishes the proof of (5.3.2).

Next, we will verify the weak inequality (5.3.3). To this end, we use Theo-rem 3.6.4 and prove that

sup�>0

�2=.2C˛/�.�1;�� a > �/ � C

for all cube 2=.2C ˛/-atom a. In other words, we have to show that

�2=.2C˛/��

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�


ˇ

ˇ

ˇ

ˇ

> �

�

� C

for i D 1; 2; 3; 4; 5 and � > 0. Since

�2=.2C˛/�.jgj > �/ �Z

jgj2=.2C˛/;

the desired inequality follows from the above inequalities for i D 1; 5.For i D 2 and p D 2=.2C ˛/, we have seen in (5.3.8) that

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A2.x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

� Cp2K.2C˛/�K�Kˇ1f2�KC4<xg1f�2�K�1<y�2�KC3g.x � 2�KC3/�1�ˇ:

If this is greater than �, then

1f2�KC4<xg.x � 2�KC3/ � C�� 11Cˇ 2

KCK˛�Kˇ1Cˇ 1f�2�K�1<y�2�KC3g:

Choosing ˇ D ˛=2, we conclude

�

�

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A2.x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

> �

�

�Z

R2

1�1

f2�KC4<xg.x�2�KC3/�C�

�2

2C˛ 2K1f�2�K�1<y�2�KC3

g

� dx dy

� C��2=.2C˛/2KZ

1f�2�K�1<y�2�KC3g dy

� C��2=.2C˛/:

If (5.3.9) (or (5.3.10)) is greater than �, then

1f2�KC1<y�x=2C2�Kg.y � 2�K�1/ � C�� 11C˛�ˇ 1f2�KC4<xg.x � 2�K�1/�

1Cˇ1C˛�ˇ :

Choosing ˇ such that � 1Cˇ1C˛�ˇ C 1 < 0, i.e. ˛=2 < ˇ � 1, we obtain

�

�

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A3 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

> �

�

�Z ��1=.2C˛/C2�K�1

2�KC4

x dx

C C�� 11C˛�ˇ

Z 1

��1=.2C˛/C2�K�1

.x � 2�K�1/�1Cˇ

1C˛�ˇ dx

� C��2=.2C˛/ C C�� 11C˛�ˇ �

�12C˛ .� 1Cˇ

1C˛�ˇ C1/

D C��2=.2C˛/:

Finally, we obtain from (5.3.11) that

1fx=2�2�K<y�x�2�KC1g.x � y � 2�K/

� C�� 11Cˇ 1f2�KC4<x<�C2�K�1g.x � 2�K�1/

ˇ�˛�11Cˇ :

Hence

�

�

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A4.x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

> �

�

�Z ��1=.2C˛/C2�K�1

2�KC4

x dx C C�� 11Cˇ

Z 1

��1=.2C˛/C2�K�1

.x � 2�K�1/ˇ�˛�11Cˇ dx

� C��2=.2C˛/ C C�� 11Cˇ �

�12C˛ .

ˇ�˛�11Cˇ C1/

D C��2=.2C˛/:

Here we have chosen ˇ such that ˇ�˛�11Cˇ C 1 < 0, i.e. 0 < ˇ � ˛=2. This finishes

the proof of the theorem. �


Proof of (5.3.2) for q D 1 and d � 3 We may assume again that ˛ � 1. Toprove (5.3.2), we will show that

Z

Rdn27I

ˇ

ˇ�1;�� a.x/ˇ

ˇ

pdx D

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp (5.3.12)

for every cube p-atom a, where ddC˛ < p < 1 and I is the support of the atom.

Without loss of generality we may suppose that a is a cube p-atom with supportI D I1 � � � Id and

Œ�2�K�2; 2�K�2� � Ij � Œ�2�K�1; 2�K�1� . j D 1; : : : ; d/ (5.3.13)

for some K 2 Z. By symmetry, we can assume that

x1 � u1 > x2 � u2 > � � � > xd � ud > 0:

Here we modify slightly the definition of the sets S and S.il; jl/;k. Let

S WD ˚

x 2 Rd W x1 > x2 > : : : > xd > 0; x1 > 2

�KC4� ;

S.il; jl/;k WD ˚

x 2 S W xil � xjl > 2�KC2; l D 1; : : : ; k � 1;

xik � xjk � 2�KC2�;

S.il; jl/;d WD ˚

x 2 S W xil � xjl > 2�KC2; l D 1; : : : ; d � 1

�

;

whenever k < d. Moreover, let

S.il; jl/;k;1 WD�

x 2 S.il; jl/;k W xjk > 2�KC2; if k < dI

x 2 S.il; jl/;k W xjd�1 > 2�KC2; if k D d;

S.il; jl/;k;2 WD�

x 2 S.il; jl/;k W xjk � 2�KC2; if k < dIx 2 S.il; jl/;k W xjd�1 � 2�KC2; if k D d:

Instead of (5.3.12), it is enough to prove

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1S.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp:

We have

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1S.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx (5.3.14)

�X

.il; jl/2I

dX

kD1

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;.il; jl/.x � u/

1S.il; jl/;k;1 .x � u/

C 1S.il; jl/;k;2 .x � u/�

du

ˇ

ˇ

ˇ

ˇ

ˇ

p

dx:


Step 1. In this step we estimate the first d � 1 summands on the set S.il; jl/;k;1 by

Cp2Kd

X

.il; jl/2I

d�1X

kD1

Z

Rdn27IsupT>0

�Z

I

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x � u/ˇ

ˇ

ˇ 1S.il; jl/;k;1 .x � u/ du

�p

dx

� Cp2Kd

3X

mD1

X

.il; jl/2I

d�1X

kD1Z

Rdn27I

�Z

IK1;�.il; jl/;m

.x � u/1S.il; jl/;k;1.x � u/ du

�p

dx:

Step 1.1. By Lemma 5.2.9 with ˇl D ˇ,

Z

IK1;�.il; jl/;1

.x � u/1S.il; jl/;k;1 .x � u/ du

� CZ

I

d�1Y

lD1.xil � uil � .xjl � ujl//

�1�ˇ.xjd�1 � ujd�1 /ˇ.d�1/�˛�1

.xid�1 � uid�1 � .xjd�1 � ujd�1 //˛1S.il; jl/;k;1.x � u/ du

� CZ

I

k�1Y


�1�ˇ

d�1Y

lDk

.xil � uil � .xjl � ujl//�1�ˇC˛=.d�k/

.xjd�1 � ujd�1 /ˇ.d�1/�˛�11S.il; jl/;k;1 .x � u/ du:

In the first product we estimate the factors and in the second one we integrate.More exactly,

xil � uil � .xjl � ujl/ > xil � xjl � 2�K ; l D 1; : : : ; k � 1;

and

xjd�1 � ujd�1 > xjd�1 � 2�K�1:

For the integration first we choose the index id�1 .D i0d�1/ and then id�2 if id�2 ¤id�1 or jd�2 if jd�2 ¤ jd�1. Repeating this process we get an injective sequence.i0l; l D k; : : : ; d � 1/. We integrate the term

.xik � uik � .xjk � ujk//�1�ˇC˛=.d�k/ with respect to ui0k

;

the term

.xikC1� uikC1

� .xjkC1� ujkC1

//�1�ˇC˛=.d�k/ with respect to ui0kC1;

. . . , and finally the term

.xid�1 � uid�1 � .xjd�1 � ujd�1 //�1�ˇC˛=.d�k/ with respect to ui0d�1

:

Since

xil � uil � .xjl � ujl/ � 2�KC2 .l D k; : : : ; d � 1/

and we can choose ˇ such that �ˇ C ˛=.d � k/ > 0, we have

Z

Il

.xil � uil � .xjl � ujl//�1�ˇC˛=.d�k/1S.il; jl/;k;1 .x � u/ duil.or dujl/

� C2�K.˛=.d�k/�ˇ/;

.l D k; : : : ; d � 1/. If x � u 2 S.il; jl/;k;1, then

xil � xjl > 2�KC2 C uil � ujl > 2

�KC2 � 2�K > 2�KC1

for all l D 1; : : : ; k � 1 and

xjd�1 > 2�KC2 C ujd�1 > 2

�KC2 � 2�K�1 > 2�KC1:

Moreover,

xil � xjl � 2�KC2 C uil � ujl < 2�KC3; l D k; : : : ; d � 1;

and

xil � xjl > uil � ujl > �2�K ; l D k; : : : ; d � 1:

HenceZ

IK1;�.il; jl/;1

.x � u/1S.il; jl/;k;1 .x � u/ du (5.3.15)

� C2�Kk2�K.˛=.d�k/�ˇ/.d�k/k�1Y

lD1.xil � xjl � 2�K/�1�ˇ1fxil�xjl>2

�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g.xjd�1 � 2�K�1/ˇ.d�1/�˛�11fxjd�1>2

�KC1g:

Consequently,

Cp2Kd

X

.il; jl/2I

d�1X

kD1

Z

Rdn27I

Z

IK1;�.il; jl/;1

.x � u/1S.il; jl/;k;1 .x � u/ du�p

dx

� Cp2Kd2�Kkp2�K.˛�ˇ.d�k//p

X

.il; jl/2I

d�1X

kD1Z

Rdn27I

k�1Y

lD1.xil � xjl � 2�K/�.1Cˇ/p1fxil �xjl>2

�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g.xjd�1 � 2�K�1/.ˇ.d�1/�˛�1/p1fxjd�1>2

�KC1g dx

� Cp2Kd2�Kkp2�K.˛�ˇ.d�k//p

X

.il; jl/2I

d�1X

kD1

2�K.1�.1Cˇ/p/.k�1/2�K.d�k/2�K.1�.˛C1�ˇ.d�1//p/

� Cp;

whenever 1 � .1 C ˇ/p < 0, 1 � .˛ C 1 � ˇ.d � 1//p < 0 and ˇ < ˛=.d � 1/.Choosing ˇ D ˛=d we obtain

p >d

d C ˛:

Step 1.2. We have

Z

IK1;�.il; jl/;2

.x � u/1S.il; jl/;k;1 .x � u/ du

� CZ

I

d�2Y


�1�ˇ.xjd�1 � ujd�1 /ˇ.d�2/�˛�1

.xid�1 � uid�1 � .xjd�1 � ujd�1 //˛�11S.il; jl/;k;1 .x � u/ du

� CZ

I

k�1Y


�1�ˇ.xjd�1 � ujd�1 /ˇ.d�2/�˛�1

d�2Y

lDk

.xil � uil � .xjl � ujl//�1�ˇC˛=.d�k/

.xid�1 � uid�1 � .xjd�1 � ujd�1 //˛=.d�k/�11S.il; jl/;k;1 .x � u/ du:

Estimating the factors in the first product and integrating the factors in the secondone, we conclude

Z

IK1;�.il; jl/;2

.x � u/1S.il; jl/;k;1 .x � u/ du

� C2�Kk2�K.˛=.d�k/�ˇ/.d�k�1/2�K˛=.d�k/

k�1Y

lD1.xil � xjl � 2�K/�1�ˇ1fxil �xjl>2

�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g.xjd�1 � 2�K�1/ˇ.d�2/�˛�11fxjd�1>2

�KC1g

and

Cp2Kd

X

.il; jl/2I

d�1X

kD1

Z

Rdn27I

�Z

IK1;�.il; jl/;2

.x � u/1S.il; jl/;k;1 .x � u/ du

�p

dx

� Cp2Kd2�Kkp2�K.˛�ˇ.d�k�1//p X

.il; jl/2I

d�1X

kD1Z

Rdn27I

k�1Y


�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g.xjd�1 � 2�K�1/.ˇ.d�2/�˛�1/p1fxjd>2

�KC1g dx

� Cp2Kd2�Kkp2�K.˛�ˇ.d�k�1//p

X

.il; jl/2I

d�1X

kD12�K.1�.1Cˇ/p/.k�1/2�K.d�k/2�K.1�.˛C1�ˇ.d�2//p/

� Cp;

whenever 1� .1Cˇ/p < 0, 1� .˛C 1�ˇ.d � 2//p < 0 and ˇ < ˛=.d � 1/. Sinceˇ can be arbitrarily near to ˛=.d � 1/, we obtain

p >d � 1

d � 1C ˛:


Step 1.3. The kernel K1;�.il; jl/;3

is the same as K1;�.il; jl/;2

with ˛ D 1. Hence

Cp2Kd

X

.il; jl/2I

d�1X

kD1

Z

Rdn27I

�Z

IK1;�.il; jl/;3

.x � u/1S.il; jl/;k;1.x � u/ du

�p

dx � Cp

as in Step 1.2, whenever

p >d � 1

d:

Step 2. Now we consider the dth summand and the set S.il; jl/;d;1, which meansthat

xil � xjl > 2�KC1 for all l D 1; : : : ; d � 1 and xjd�1 > 2

�KC1:

Now we split the supremum into two parts:

X

.il; jl/2I

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;.il; jl/.x � u/1S.il; jl/;d;1 .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

�X

.il; jl/2I

Z

Rdn27I

supT>2K

C supT�2K

!

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;.il; jl/.x � u/1S.il; jl/;d;1.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx: (5.3.16)

Step 2.1. We use Lemma 5.2.7 with ˇl D ˇ and similar methods as in Step 1 toobtain

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

� Cp2Kd=p2�K˛

Z

I

d�1Y


�1�ˇ

.xjd�1 � ujd�1 /ˇ.d�1/�˛�11S.il; jl/;d;1.x � u/ du

� Cp2Kd=p2�K˛2�Kd

d�1Y


�KC1g

.xjd�1 � 2�K�1/ˇ.d�1/�˛�11fxjd�1>2�KC1g

and

X

.il; jl/2I

Z

Rdn27Isup

T>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd2�Kdp2�K˛p

X

.il; jl/2I

Z

Rdn27I

d�1Y

lD1.xil � xjl � 2�K/�.1Cˇ/p

1fxil�xjl>2�KC1g.xjd�1 � 2�K�1/.ˇ.d�1/�˛�1/p1fxjd�1>2

�KC1g dx

� Cp2Kd2�Kdp2�K˛p

X

.il; jl/2I2�K.1�.1Cˇ/p/.d�1/2�K.1�.˛C1�ˇ.d�1//p/

� Cp;

whenever 1 � .1C ˇ/p < 0 and 1 � .˛ C 1 � ˇ.d � 1//p < 0. That is to say (forˇ D ˛=d)

p >d

d C ˛:

Step 2.2. We may suppose that the centre of I is zero, in other words I WDQd

jD1.��; �/ for some � > 0. Let

A1.u/ WDZ u1

��a.t1; u2; : : : ; ud/ dt1

and

Ak.u/ WDZ uk

��Ak�1.u1; : : : ; uk�1; tk; ukC1; : : : ; ud/ dtk; .2 � k � d/:

Observe that

jAk.u/j � Cp2K.d=p�k/:

Integrating by parts, we can see that

Z

I1

a.u/K1;�T;.il; jl/

.x � u/1S.il; jl/;d;1.x � u/ du1

D A1.�; u2; : : : ; ud/.K1;�T;.il ; jl/

1S.il; jl/;d;1 /.x1 � �; x2 � u2; : : : ; xd � ud/

CZ �

��A1.u/@1K

1;�T;.il ; jl/

.x � u/1S.il; jl/;d;1 .x � u/ du1;


because A1.��; u2; : : : ; ud/ D 0. Integrating the first term again by parts, we obtain

Z

I1

Z

I2

a.u/K1;�T;.il; jl/

.x � u/1S.il; jl/;d;1.x � u/ du1du2

D A2.�; �; u3; : : : ; ud/

.K1;�T;.il ; jl/

1S.il; jl/;d;1 /.x1 � �; x2 � �; x3 � u3; : : : ; xd � ud/

CZ �

��A2.�; u2; : : : ; ud/

.@2K1;�T;.il ; jl/

1S.il; jl/;d;1/.x1 � �; x2 � u2; : : : ; xd � ud/ du2

CZ

I1

Z

I2

A1.u/.@1K1;�T;.il ; jl/

1S.il; jl/;d;1/.x � u/ du1du2:

We repeat this process to get that

Z

Ia.u/K1;�

T;.il; jl/.x � u/1S.il; jl/;d;1.x � u/ du (5.3.17)

DdX

kD1

Z

Ik

: : :

Z

Id

Ak.u.k//.@kK1;�

T;.il ; jl/1S.il; jl/;d;1 /.x � u.k// duk : : : dud;

where u.k/ WD .�; : : : ; �; uk; : : : ; ud/. Remark that Ad.�; : : : ; �/ D R

I a D 0.Lemma 5.2.10 with ˇl D ˇ and the last inequality imply

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

� Cp

dX

kD12K.d=p�k/ (5.3.18)

Z

Ik

: : :

Z

Id

k�1Y

lD1.xil � � � .xjl � �//�1�ˇ

d�1Y

lDk

.xil � uil � .xjl � ujl//�1�ˇ

.xjd�1 � ujd�1 /ˇ.d�1/�˛�12K.1�˛/ C .xjd�1 � ujd�1 /

ˇ.d�1/�2�

1S.il; jl/;d;1.x � u/ duk : : : dud

� Cp2K.d=p�k/2�K.d�kC1/

d�1Y


�KC1g

.xjd�1 � 2�K�1/ˇ.d�1/�˛�12K.1�˛/ C .xjd�1 � 2�K�1/ˇ.d�1/�2�

1fxjd�1>2�KC1g

and

X

.il; jl/2I

Z

Rdn27Isup

T�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd2�Kdp�Kp

X

.il; jl/2I

Z

Rdn27I

d�1Y

lD1.xil � xjl � 2�K/�.1Cˇ/p1fxil�xjl>2

�KC1g1fxjd�1>2�KC1g

.xjd�1 � 2�K�1/.ˇ.d�1/�˛�1/p2K.1�˛/p C .xjd�1 � 2�K�1/.ˇ.d�1/�2/p�

dx

� Cp2Kd2�Kdp�Kp

X

.il; jl/2I2�K.1�.1Cˇ/p/.d�1/

2�K.1�.˛C1�ˇ.d�1//p/2K.1�˛/p C 2�K.1�.2�ˇ.d�1//p/�

� Cp; (5.3.19)

whenever p > ddC˛ .

Step 3. Now we investigate the first d � 1 summands and the set S.il; jl/;k;2in (5.3.14). In this case

xik � uik � .xjk � ujk/ � 2�KC2 and xjd�1 < xik < 2�KC4:

Note that xjd�1 � 2�KC2 for k D d. Observe that k D 1 can be excluded, becausei1 D 1, j1 D d and this contradicts the definition of S, where x1 > 2�KC4.

Step 3.1. Similar to (5.3.15), we can conclude that

Z

IK1;�.il; jl/;1

.x � u/1S.il; jl/;k;2.x � u/ du

� C2�K.k�1/2�K.˛=.d�k/�ˇ/.d�k/2�K.ˇ.d�1/�˛/

k�1Y


�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g1fxjd�1<2

�KC4g

and

Cp2Kd

X

.il; jl/2I

d�1X

kD2

Z

Rdn27I

�Z

IK1;�.il; jl/;1

.x � u/1S.il; jl/;k;2 .x � u/ du

�p

dx

� Cp2Kd2�K.k�1/p2�K.˛�ˇ.d�k//p2�K.ˇ.d�1/�˛/p X

.il; jl/2I

d�1X

kD1

Z

Rdn27I

k�1Y


�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g1fxjd�1<2

�KC4g dx

� Cp2Kd2�K.k�1/p2�K.˛�ˇ.d�k//p2�K.ˇ.d�1/�˛/p

X

.il; jl/2I

d�1X

kD12�K.1�.1Cˇ/p/.k�1/2�K.d�kC1/

� Cp;

whenever 1�.1Cˇ/p < 0, ˇ.d �1/�˛ > 0 and ˇ < ˛=.d �k/ (k D 2; : : : ; d �1).Since ˇ can be near to ˛=.d � 2/ we obtain

p >d � 2

d � 2C ˛:

Step 3.2. Using the method of Step 1.2, we get that

Z

IK1;�.il; jl/;2

.x � u/1S.il; jl/;k;2.x � u/ du

� CZ

I

d�2Y


�1�ˇ.xjd�1 � ujd�1 /ˇ.d�2/�˛�1

.xid�1 � uid�1 � .xjd�1 � ujd�1 //˛�11S.il; jl/;k;2 .x � u/ du

� CZ

I

k�1Y


�1�ˇ1S.il; jl/;k;2 .x � u/

d�2Y

lDk

.xil � uil � .xjl � ujl//�1�ˇC.˛��/=.d�k�1/

.xid�1 � uid�1 � .xjd�1 � ujd�1 //��1.xjd�1 � ujd�1 /

ˇ.d�2/�˛�1 du

� C2�K.k�1/2�K..˛��/=.d�k�1/�ˇ/.d�k�1/2�K�

k�1Y


�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g2�K.ˇ.d�2/�˛/1fxjd�1�2�KC4g

and so

Cp2Kd

X

.il; jl/2I

d�2X

kD2

Z

Rdn27I

�Z

IK1;�.il; jl/;2

.x � u/1S.il; jl/;k;2.x � u/ du

�p

dx

� Cp2Kd2�K.k�1/p2�K.˛�ˇ.d�k�1//p2�K.ˇ.d�2/�˛/p X

.il; jl/2I

d�1X

kD2Z

Rdn27I

k�1Y


�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g1fxjd �2�KC4g dx

� Cp2Kd2�K.k�1/p2�K.˛�ˇ.d�k�1//p2�K.ˇ.d�2/�˛/p

X

.il; jl/2I

d�1X

kD12�K.1�.1Cˇ/p/.k�1/2�K.d�kC1/

� Cp:

We have used that 0 < � < ˛, ˛=.d�2/ < ˇ < .˛��/=.d�3/ and 1�.1Cˇ/p < 0.(Here d > 3.) This implies that � < ˛=.d � 2/. Since ˇ can be chosen arbitrarilynear to .˛ � �/=.d � 3/ and � to 0, we obtain

p >d � 3

d � 3C ˛:

Observe that the same estimation is true if k D d � 1 with p > d�2d�2C˛ .

Step 3.3. As we mentioned before the kernel K1;�.il; jl/;3

is the same as K1;�.il ; jl/;2

with˛ D 1.

Step 4. Here we consider the dth summand and the set S.il; jl/;d;2 in (5.3.14).

Step 4.1. Since xil � xjl � 2�KC2, we get as in Step 2.1 that

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

� Cp2Kd=p2�K˛2�K.d�1/

d�1Y

lD1.xil � xjl � 2�K/�1�ˇ1fxil�xjl>2

�KC1g

2�K.ˇ.d�1/�˛/1fxjd�1�2�KC4g

and

X

.il; jl/2I

Z

Rdn27Isup

T>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd2�K.d�1/p2�K˛p2�K.ˇ.d�1/�˛/p X

.il; jl/2I

Z

Rdn27I

d�1Y


�KC1g1fxjd�1�2�KC1g dx

� Cp2Kd2�K.d�1/p2�K˛p2�K.ˇ.d�1/�˛/p X

.il; jl/2I2�K.1�.1Cˇ/p/.d�1/2�K

� Cp;

whenever 1 � .1C ˇ/p < 0 and ˇ.d � 1/� ˛ > 0. In other words p > ddC˛ .

Step 4.2. Similar to (5.3.18),

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

� Cp

dX

kD12K.d=p�k/

Z

Ik

: : :

Z

Id

k�1Y

lD1.xil � � � .xjl � �//�1�ˇ

d�1Y

lDk

.xil � uil � .xjl � ujl//�1�ˇ

.xjd�1 � ujd�1 /ˇ.d�1/�˛�12K.1�˛/ C .xjd�1 � ujd�1 /

ˇ.d�1/�2�

1S.il; jl/;d;2.x � u/ duk : : : dud

� Cp2K.d=p�k/2�K.d�k/

d�1Y


�KC1g

2�K.ˇ.d�1/�˛/2K.1�˛/ C 2�K.ˇ.d�1/�1/�1fxjd�1�2�KC4g;


thus

X

.il; jl/2I

Z

Rdn27Isup

T�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;.il ; jl/.x � u/1S.il; jl/;d;2 .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd2�Kdp

X

.il; jl/2I

Z

Rdn27I

d�1Y


�KC1g

2�K.ˇ.d�1/�˛/p2K.1�˛/p C 2�K.ˇ.d�1/�1/p�

1fxjd�1�2�KC4g dx

� Cp2Kd2�Kdp

X

.il; jl/2I2�K.1�.1Cˇ/p/.d�1/

2�K.ˇ.d�1/�˛/p2K.1�˛/p C 2�K.ˇ.d�1/�1/p�2�K

� Cp;

whenever p > ddC˛ . This finishes the proof of (5.3.2). �

Proof of (5.3.3) for q D 1 and d � 3 By Theorem 3.6.4, to prove the weak inequal-ity (5.3.3), it is enough to show that

sup�>0

�d=.dC˛/��

�1;�� a > �;Rd n 27I� � C (5.3.20)

for all cube d=.d C ˛/-atoms a, where I is the support of the atom and ˛ � 1. Wemay suppose again that I D I1 : : : Id and

Œ�2�K�2; 2�K�2� � Ij � Œ�2�K�1; 2�K�1� . j D 1; : : : ; d/

for some K 2 Z. It is easy to see that instead of (5.3.20) it is enough to prove that

�d=.dC˛/��

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;.il; jl/.x � u/1S.il; jl/;k;l.x � u/ du

ˇ

ˇ

ˇ

ˇ

> �;Rd n 27I�

� C (5.3.21)

for all � > 0, 1 � k � d and l D 1; 2. We have proved above that

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;.il; jl/.x � u/1S.il; jl/;k;l.x � u/ du

ˇ

ˇ

ˇ

ˇ

d=.dC˛/dx � C (5.3.22)

for 1 � k � d � 1 and l D 2. Inequality

�d=.dC˛/�.jgj > �/ �Z

jgjd=.dC˛/

implies (5.3.21) for these parameters.


Step 5. Let 1 � k � d � 1 and l D 1. By Lemma 5.2.9, we have to show that

�d=.dC˛/��Z

Ija.u/jK1;�

.il; jl/;m.x � u/1S.il; jl/;k;1.x � u/ du > �;Rd n 27I

�

� C

for 1 � k � d�1 and m D 1; 2; 3. This is known for m D 2; 3, because the analogueof (5.3.22) was proved above. So assume that m D 1. Let us apply Lemma 5.2.9 with

ˇ1 D ˛=d C �; � � 0 and ˇl D ˇ; l D 2; : : : ; d � 1:

If x 2 S.il; jl/;k;1, then

K1;�.il; jl/;1

.x/

� C.xi1 � xj1 /� dC˛

d ��d�1Y

lD2.xil � xjl/

�1�ˇxˇ.d�2/C ˛

d C��˛�1jd�1

.xid�1 � xjd�1 /˛

� C.xi1 � xj1 /� dC˛

d

k�1Y

lD2.xil � xjl/

�1�ˇ� �k�2

d�1Y

lDk

.xil � xjl/�1�ˇC ˛

d�k xˇ.d�2/C ˛

d C��˛�1jd�1

:

If k D 1; 2, then the first product disappears and let � D 0. In all cases

Z

Ija.u/jK1;�

.il; jl/;1.x � u/1S.il; jl/;k;1 .x � u/ du

� C2K.dC˛/Z

I.xi1 � ui1 � .xj1 � uj1/

� dC˛d

k�1Y

lD2.xil � uil � .xjl � ujl/

�1�ˇ� �k�2

d�1Y

lDk

.xil � uil � .xjl � ujl/�1�ˇC ˛

d�k

.xjd�1 � ujd�1 /ˇ.d�2/C ˛

d C��˛�11S.il; jl/;k;1 .x � u/ du:

Similar to (5.3.15),

Z

Ija.u/jK1;�

.il; jl/;1.x � u/1S.il; jl/;k;1.x � u/ du

� C2K.dC˛/2�Kk2�K.˛=.d�k/�ˇ/.d�k/.xi1 � xj1 � 2�K/�dC˛

d

k�1Y

lD2.xil � xjl � 2�K/�1�ˇ� �

k�2 1fxil �xjl>2�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g.xjd�1 � 2�K�1/ˇ.d�2/C ˛

d C��˛�1

1fxi1�xj1>2�KC1g1fxjd�1>2

�KC1g:

If this is greater than � then

.xi1 � xj1 � 2�K/1fxi1�xj1>2�KC1g

� C�� ddC˛ 2Kd2� Kkd

dC˛ 2�K.˛�ˇ.d�k// ddC˛ 1fxjd�1>2

�KC1gk�1Y

lD2.xil � xjl � 2�K/.�1�ˇ� �

k�2 /d

dC˛ 1fxil �xjl>2�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g.xjd�1 � 2�K�1/.ˇ.d�2/C ˛

d C��˛�1/ ddC˛ :

Note that either i1 > i2 or j1 < j2. Suppose that the first one is the case, then letx0 WD .x1; : : : ; xi1�1; ; xi1C1; : : : ; xd/. Thus

�

Z

Ija.u/jK1;�

.il; jl/;1.x � u/1S.il; jl/;k;1 .x � u/ du > �;Rd n 27I

�


dC˛ 2�K.˛�ˇ.d�k// ddC˛

Z k�1Y

lD2.xil � xjl � 2�K/.�1�ˇ� �

k�2 /d

dC˛ 1fxil �xjl>2�KC1g

d�1Y

lDk

1f�2�K<xil �xjl<2�KC3g

.xjd�1 � 2�K�1/.ˇ.d�2/C ˛d C��˛�1/ d

dC˛ 1fxjd�1>2�KC1g dx0


dC˛ 2�K.˛�ˇ.d�k// ddC˛ 2�K..�1�ˇ� �

k�2 /d

dC˛C1/.k�2/

2�K.d�k/2�K..ˇ.d�2/C ˛d C��˛�1/ d

dC˛C1/

D C�� ddC˛ ;

whenever

d C ˛ < .1C ˇ C �

k � 2/d; d C ˛ < .�ˇ.d � 2/� ˛

d� � C ˛ C 1/d;

in other words

˛

d� �

k � 2< ˇ <

˛

d� �

d � 2:

Note that if k D 1; 2, then � D 0 and the left-hand side disappears. For other k’s wecan choose a small � > 0 such that ˇ satisfies all inequalities mentioned above.

Step 6. Suppose that k D d and l D 1. Taking into account (5.3.16), we get twosubsteps.

Step 6.1. We use Lemma 5.2.7 to obtain

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

� C2K.dC˛/2�K˛Z

I

d�1Y


�1�ˇl

.xjd�1 � ujd�1 /Pd�1

lD1 ˇl�˛�11S.il; jl/;d;1 .x � u/ du

� Cd�1Y

lD1.xil � xjl � 2�K/�1�ˇl1fxil �xjl>2

�KC1g

.xjd�1 � 2�K�1/Pd�1

lD1 ˇl�˛�11fxjd�1>2�KC1g: (5.3.23)

If the right-hand side is greater than �, then

.xi1 � xj1 � 2�K/1fxi1�xj1>2�KC1g

� C��11Cˇ1

d�1Y

lD2.xil � xjl � 2�K/

�1�ˇl1Cˇ1 1fxil �xjl>2

�KC1g

.xjd�1 � 2�K�1/Pd�1

lD1 ˇl�˛�1

1Cˇ1 1fxjd�1>2�KC1g:

For k D 2; : : : ; d � 1, let

Rk WD(

x0 W xil � xjl � 2�K > ��1

dC˛ ; l D 2; : : : ; k � 1Ix0 W xil � xjl � 2�K � �

�1dC˛ ; l D k; : : : ; d � 1

and

Rk;1 WD fx0 2 Rk W xjd�1 � 2�KC1 > ��1

dC˛ g;Rk;2 WD fx0 2 Rk W xjd�1 � 2�KC1 � �

�1dC˛ g:

We may assume that x0 2 Rk;1 or x0 2 Rk;2. In both cases let ˇ2 D ˇ3 D : : : D ˇk�1and ˇk D ˇkC1 D : : : D ˇd�1. Then

�

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

1Rk;1.x0/ > �

!

� C��11Cˇ1

Z k�1Y

lD2.xil � xjl � 2�K/

�1�ˇ21Cˇ1 1fxil �xjl>2

�KC1g

d�1Y

lDk

.xil � xjl � 2�K/�1�ˇk1Cˇ1 1fxil �xjl>2

�KC1g

.xjd�1 � 2�K�1/ˇ1Cˇ2.k�2/Cˇk.d�k/�˛�1

1Cˇ1 1fxjd�1>2�KC1g1Rk;1 .x

0/ dx0

� C��11Cˇ1 �

�1dC˛ .

�1�ˇ21Cˇ1

C1/.k�2/�

�1dC˛ .

�1�ˇk1Cˇ1

C1/.d�k/

��1

dC˛ .ˇ1Cˇ2.k�2/Cˇk.d�k/�˛�1

1Cˇ1C1/

� C��d

dC˛ ; (5.3.24)

whenever

1C ˇk < 1C ˇ1 < 1C ˇ2; 1C ˇ1 < ˛ C 1 � ˇ1 � ˇ2.k � 2/� ˇk.d � k/:

Substituting ˇi D ˛=d C i .i D 1; 2; k/ with small i in these inequalities, weobtain

1 < 2; 1 > k; 0 < �2 1 � 2.k � 2/� k.d � k/:

If 1 < 0, 2 > 0 and k < 0 are small enough, then these inequalities are satisfiedfor a fixed k.

For the set Rk;2 we get the same inequality as in (5.3.24) if

1C ˇk < 1C ˇ1 < 1C ˇ2; 0 < ˛ C 1 � ˇ1 � ˇ2.k � 2/� ˇk.d � k/ < 1C ˇ1;

or, after the substitution,

1 < 2; 1 > k; �1d

� 1 � 1 < �2 1 � 2.k � 2/� k.d � k/ < 0:

These inequalities are again satisfied if the absolute values of 1 > 0, 2 > 0 and k < 0 are small enough.

Step 6.2. Assume that T � 2K . Similar to (5.3.18), Lemma 5.2.10 and theinequality (5.3.17) imply

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

� CdX

kD12K.dC˛�k/

Z

Ik

: : :

Z

Id

k�1Y

lD1.xil � � � .xjl � �//�1�ˇl

d�1Y

lDk

.xil � uil � .xjl � ujl//�1�ˇl

.xjd�1 � ujd�1 /Pd�1

lD1 ˇl�˛�12K.1�˛/ C .xjd�1 � ujd�1 /Pd�1

lD1 ˇl�2�

1S.il; jl/;d;1 .x � u/ duk : : : dud

DW A.x/C B.x/; (5.3.25)

where A.x/ denotes the expression with .xjd�1 � ujd�1 /Pd�1

lD1 ˇl�˛�12K.1�˛/ and B.x/

with .xjd�1 � ujd�1 /Pd�1

lD1 ˇl�2. If ˛ D 1, then these two summands are the same. If˛ < 1, then we can show for the second summand as in (5.3.19) that

Z

Rdn27IjB.x/jp dx � C (5.3.26)

for p D ddC˛ . We obtain for A.x/ that

A.x/ � C2K.dC˛�k/d�1Y

lD1.xil � xjl � 2�K/�1�ˇl1fxil �xjl>2

�KC1g

2�K.d�kC1/2K.1�˛/.xjd�1 � 2�K�1/Pd�1

lD1 ˇl�˛�11fxjd�1>2�KC1g:

Since this is the same as (5.3.23), the proof can be finished as in Step 6.1.Step 7. Here we consider k D d and l D 2.Step 7.1. Let T > 2K . Setting ˇ1 D ˛=d C �, � > 0 and ˇl D ˇ, l D 2; : : : ; d � 1

in Lemma 5.2.7, we obtain

ˇ

ˇ

ˇK1;�T;.il ; jl/

.x/ˇ

ˇ

ˇ � CT�˛.xi1 � xj1 /� dC˛

d

d�1Y

lD2.xil � xjl/

�1�ˇ� �d�2 x

ˇ.d�2/C ˛d C��˛�1

jd�1:

Since xil � xjl � 2�KC2, we get that

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

(5.3.27)

� C2K.dC˛/2�K˛2�K.d�1/.xi1 � xj1 � 2�K/�dC˛

d 1fxi1�xj1>2�KC1g

d�1Y

lD2.xil � xjl � 2�K/�1�ˇ� �

d�2 1fxil�xjl>2�KC1g

2�K.ˇ.d�2/C ˛d C��˛/1fxjd�1�2�KC4g:

If this is greater than �, then

.xi1 � xj1 � 2�K/1fxi1�xj1>2�KC1g

� C�� ddC˛ 2�K.ˇ.d�2/C ˛

d C��˛�1/ ddC˛

d�1Y

lD2.xil � xjl � 2�K/.�1�ˇ� �

d�2 /d

dC˛ 1fxil �xjl>2�KC1g1fxjd�1�2�KC1g:

We conclude that

�

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�


ˇ

ˇ

ˇ

ˇ

> �;Rd n 27I!

� C�� ddC˛ 2�K.ˇ.d�2/C ˛

d C��˛�1/ ddC˛ 1fxjd�1�2�KC1g

Z d�1Y

lD2.xil � xjl � 2�K/.�1�ˇ� �

d�2 /d

dC˛ 1fxil �xjl>2�KC1g dx0

� C�� ddC˛ 2�K.ˇ.d�2/C ˛

d C��˛�1/ ddC˛ 2�K..�1�ˇ� �

d�2 /d

dC˛C1/.d�2/2�K

D C�� ddC˛ ;

whenever

0 < ˇ.d � 2/C ˛

dC � � ˛ < 1; d C ˛ < d

1C ˇ C �

d � 2

�

;

in other words

˛

d� �

d � 2 _ ˛ � � � ˛d

d � 2 < ˇ <˛ C 1 � � � ˛

d

d � 2:

Step 7.2. Now let T � 2K . Let us consider (5.3.25) with the set S.il; jl/;d;2 insteadof S.il; jl/;d;1. Then (5.3.26) holds for ˛ < 1. In the expression of A let us writeˇ1 D ˛=d C �, � > 0 and ˇl D ˇ, l D 2; : : : ; d � 1 to obtain

A.x/

� CdX

kD12K.dC1�k/

Z

Ik

: : :

Z

Id

.xi1 � ui1 � .xj1 � uj1 //� dC˛

d

k�1Y

lD2.xil � � � .xjl � �//�1�ˇ� �

d�2

d�1Y

lDk

.xil � uil � .xjl � ujl//�1�ˇ� �

d�2

.xjd�1 � ujd�1 /ˇ.d�2/C ˛

d C��˛�11S.il; jl/;d;2 .x � u/ duk : : : dud

� CdX

kD12K.dC1�k/2�K.d�k/2�K.ˇ.d�2/C ˛

d C��˛/.xi1 � xj1 � 2�K/�dC˛

d

1fxi1�xj1>2�KC1g

d�1Y

lD2.xil � xjl � 2�K/�1�ˇ� �

d�2 1fxil �xjl>2�KC1g

1fxjd�1�2�KC4g;

which is the same as (5.3.27). Note that ui1 D uj1 D � if k � 2. This completes theproof of (5.3.3). �


Proof of Theorem 5.3.2 for q D 1 and d D 2 We assume again that ˛ � 1 and a isa cube p-atom with support I D I1 I2,

Œ�2�K�2; 2�K�2� � Ij � Œ�2�K�1; 2�K�1� . j D 1; 2/

for some K 2 Z. As before, it is enough to show that

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�


ˇ

ˇ

ˇ

ˇ

p

dx dy � Cp

for all i D 1; 2; 3; 4; 5, where x � u > y � v > 0 and

A1 WD f.x; y/ W 0 < x � 2�KC5; 0 < y < xg;A2 WD f.x; y/ W x > 2�KC5; 0 < y � 2�KC2g;A3 WD f.x; y/ W x > 2�KC5; 2�KC2 < y � x=2g;

A4 WD f.x; y/ W x > 2�KC5; x=2 < y � x � 2�KC2g;A5 WD f.x; y/ W x > 2�KC5; x � 2�KC2 < y < xg:

The estimation on the set A1 is the same as before in the proof for q D 1.Using (5.2.12), we conclude

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A2.x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

� Cp22K=p

Z

I.x � u � y C v/�1�˛.y � v/˛�11A2.x � u; y � v/ dudv

� Cp22K=p�K˛1f2�KC4<xg1f�2�K�1<y�2�KC3g

Z

I1

.x � 2�KC3/�1�˛ du

� Cp22K=p�K�K˛1f2�KC4<xg1f�2�K�1<y�2�KC3g.x � 2�KC3/�1�˛

and

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A2.x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dx dy


Z 1

2�KC4

Z 2�KC3

�2�K�1

.x � 2�KC3/�p.1C˛/ dydx

� Cp;

provided that p > 1=.1C ˛/. By (5.2.13),

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A3 .x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

� CpT�˛22K=pZ

I.x � u/�1�˛Cˇ.y � v/�1�ˇ1A3.x � u; y � v/ dudv


.x � 2�K�1/�1�˛Cˇ.y � 2�K�1/�1�ˇ;

whenever T > 2K . On A3 the inequality

ˇ

ˇ

ˇ@jK1;�T .x; y/

ˇ

ˇ

ˇ � CT1�˛x�1�˛Cˇy�1�ˇ

follows from Lemma 5.2.12. Similar to the proof for q D 1, we get by integrationby parts that

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A3 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

� CpT1�˛22K=p�2KZ

I2

.x � �/�1�˛Cˇ.y � v/�1�ˇ1A3.x � �; y � v/ dv

C CpT1�˛22K=p�KZ

I.x � u/�1�˛Cˇ.y � v/�1�˛1A3 .x � u; y � v/ dudv


.x � 2�K�1/�1�˛Cˇ.y � 2�K�1/�1�˛

if T � 2K . ThusZ

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A3 .x � u; y � v/ du dv

ˇ

ˇ

ˇ

ˇ

p

dxdy


Z 1

2�KC4

Z x=2C2�K

2�KC1

.x � 2�K�1/.�1�˛Cˇ/p.y � 2�K�1/�.1Cˇ/p dydx

� Cp22K�2Kp�K˛p2�K.1�.˛�ˇC1/p/2�K.1�.1Cˇ/p/

� Cp;

whenever p > 1=.1C ˇ/ and p > 1=.˛ � ˇ C 1/. ˇ D ˛=2 implies p > 22C˛ :

Using (5.2.14), we see thatˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A4 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

� CpT�˛22K=p

Z

I.x � u � y C v/�1�˛Cˇ.x � u/�1�ˇ1A4.x � u; y � v/ dudv

� CpT�˛22K=p�2K1f2�KC4<xg1fx=2�2�K<y�x�2�KC1g

.x � y � 2�K/�1�˛Cˇ.x � 2�K�1/�1�ˇ

and

Z

R2

supT>2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A4 .x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

p

dxdy

Z 1

2�KC4

Z x�2�KC1

x=2�2�K.x � y � 2�K/.�1�˛Cˇ/p.x � 2�K�1/�.1Cˇ/p dydx

� Cp22K�2Kp�K˛p2�K.1�.˛�ˇC1/p/2�K.1�.1Cˇ/p/

� Cp;

whenever ˇ D ˛=2 and p > 22C˛ . The inequality

Z

R2

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A4.x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

p

dxdy � Cp

can be shown similarly. Finally,

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A5.x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

� Cp22K=p

Z

I.x � u/�21A5 .x � u; y � v/ dudv

� Cp22K=p�2K1f2�KC4<xg1fx�2�KC3<y�xC2�Kg.x � 2�K�1/�2

and so

Z

R2

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u; v/K1;�

T .x � u; y � v/1A5.x � u; y � v/ dudv

ˇ

ˇ

ˇ

ˇ

p

dxdy

� Cp22K�2Kp

Z 1

2�KC4

Z xC2�K

x�2�KC3

.x � 2�K�1/�2p dydx

� Cp:

The proof of the weak inequality (5.3.3) is similar to that for q D 1, the detailsare left to the reader. �


Proof of (5.3.2) for q D 1 and d � 3 We will show again that

Z

Rdn27I

ˇ

ˇ�1;�� a.x/ˇ

ˇ

pdx D

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp

(5.3.28)

for every cube p-atom a, where ddC˛ < p < 1 and ˛ � 1. Assume again that I, the

support of a, satisfies (5.3.13).

We modify the definition of the sets S, S�0 , S 0 and Sk as follows:

S WD ˚

x 2 Rd W x1 > x2 > � � � > xd > 0; x1 > 2

�KC5� ;

Sk WD ˚

x 2 S W x1 > x2 > � � � > xk � 2�KC2 > xkC1 > � � � > xd > 0�

;

S�0 WD8

<

:

x 2 Rd Wˇ

ˇ

ˇ

ˇ

ˇ

ˇ

d�1X

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< d2�KC49

=

;

;

S 0 WD8

<

:

x 2 Rd W 9�;

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< d2�KC49

=

;

;

k D 1; : : : ; d. The sets S�;1 and S�0 ;d are defined as before in Sect. 5.2.1.4,

S�;1 WD8

<

:

x 2 Rd Wˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< 4x1

9

=

;

;

S�0 ;d WD8

<

:

x 2 Rd Wˇ

ˇ

ˇ

ˇ

ˇ

ˇ

d�1X

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< 4xd

9

=

;

:

It is easy to see that the lemmas of Sect. 5.2.1.4 hold for these sets, too.Instead of (5.3.28) it is enough to prove by symmetry that

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1S.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx (5.3.29)

�X

kD1;d

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1Sk\S0.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

CX

�0

d�1X

kD2

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1Sk\S�0 .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

CX

kD1;d

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1SknS0.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

CX

�0

d�1X

kD2

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1SknS�0 .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

� Cp:

Step 1. Let us consider the first sum of (5.3.29). Since u 2 I, x � u 2 S�0 orx � u 2 S 0 implies that x1 must be in an interval of length C2�K . If x � u 2 Sk

and u 2 I, then xi � ui � 2�KC2 and so xi � 2�KC1 .i D 1; : : : ; k/, moreover,xi � ui < 2�KC2 and so xi < 2�KC3 .i D k C 1; : : : ; d/. By Theorem 5.2.1, theintegral of K1;�

T can be estimated by a constant, thus

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1S1\S0.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp2Kd2�Kd:

If k D d, then Lemma 5.2.13 implies

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1Sd\S0.x � u/ du

ˇ

ˇ

ˇ

ˇ

� C2Kd=pZ

I

dY

iD1.xi � ui/

�11Sd\S0.x � u/ du

� C2Kd=pZ

I

dY

iD2.xi � 2�K�1/�1�1=.d�1/1Sd\S0.x � u/ du

� C2Kd=p�Kd1I0

1.x1/

dY

iD2.xi � 2�K�1/�d=.d�1/1fxi�2�KC1g

!

;

where the length of I01 is c2�K . Then

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1Sd\S0.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd�Kdp2�K

Z

Rd�1

dY

iD2.xi � 2�K�1/�dp=.d�1/1fxi�2�KC1g

!

dx2 : : : dxd

� Cp;

when p > .d � 1/=d.Step 2. In the second sum let us investigate first the term multiplied by 1S�0;d .x�u/

in the integrand for all k D 2; : : : ; d � 1. If x � u 2 S�0 ;d, then u1 is in an interval oflength 8.xd � ud/. By Lemma 5.2.13,

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1Sk\S�0\S�0 ;d.x � u/ du

ˇ

ˇ

ˇ

ˇ

� C2Kd=pZ

I

dY

iD1.xi � ui/

�11Sk\S�0 \S�0 ;d.x � u/ du

� C2Kd=p1I0

1.x1/

Z

I2��Id

kY

iD2.xi � ui/

�1�1=.k�1/!

dY

iDkC1.xi � ui/

�1!

.xd � ud/1Sk.x � u/ du2 : : : dud

� C2Kd=p1I0

1.x1/

Z

I2��Id

kY

iD2.xi � 2�K�1/�1�1=.k�1/

!

dY

iDkC1.xi � ui/

�1C1=.d�k/

!

1Sk.x � u/ du2 : : : dud

� C2Kd=p�K.k�1/�K1I0

1.x1/

kY

iD2.xi � 2�K�1/�k=.k�1/1fxi�2�KC1g

!

dY

iDkC11fxi<2�KC3g

!

;

where the length of I01 is c2�K . Hence

d�1X

kD2

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1Sk\S�0 \S�0;d .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd�Kkp2�K2�K.�kp=.k�1/C1/.k�1/2�K.d�k/

� Cp;

whenever p > .d � 1/=d.If x � u 62 S�0 ;d then jPd

jD1 �j.xj � uj/j � 3.xd � ud/. Applying this andLemma 5.2.15, we can see that

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1Sk\S�0 nS�0 ;d .x � u/ du

ˇ

ˇ

ˇ

ˇ

� CX

�d

2Kd=pZ

I

d�1Y

iD1.xi � ui/

�1!

.xd � ud/˛�1�ı

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛Cı

1Sk\S�0 .x � u/ du

� CX

�d

2Kd=pZ

I

kY

iD2.xi � 2�K�1/�1�1=.k�1/

!

dY

iDkC1.xi � ui/

�1C.˛�ı/=.d�k/

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛Cı

1Sk\S�0 .x � u/ du

� C2Kd=p�K.k�1/�K.˛�ı/�K.�˛CıC1/1I0

1.x1/

kY

iD2.xi � 2�K�1/�k=.k�1/1fxi�2�KC1g

!

dY

iDkC11fxi<2�KC3g

!

and

d�1X

kD2

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1Sk\S�0 nS�0 ;d.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp;

as before, whenever ˛ � 1 < ı < ˛ and p > .d � 1/=d. This proves that the secondsum in (5.3.29) can be estimated by a constant.

Step 3. Now let us consider the fourth sum of (5.3.29):

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1SknS�0 .x � u/ du

ˇ

ˇ

ˇ

ˇ

� CX

�d

2Kd=pZ

I

d�1Y

iD1.xi � ui/

�1!

.xd � ud/˛�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SknS�0 /\S�;1.x � u/C 1.SknS�0 /nS�;1 .x � u/�

du;

k D 2; : : : ; d � 1. From this it follows that

X

�d

2Kd=pZ

I

d�1Y

iD1.xi � ui/

�1!

.xd � ud/˛�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SknS�0 /\S�;1.x � u/ du

� CX

�d

2Kd=pZ

I

kY

iD2.xi � ui/

�1C.ı�1/=.k�1/!

dY

iDkC1.xi � ui/

˛=.d�k/�1!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

1SknS�0 .x � u/ du

� CX

�d

2Kd=p�Kk�K˛

kY

iD2.xi � 2�K�1/�.k�ı/=.k�1/1fxi�2�KC1g

!

dY

iDkC11fxi<2�KC3g

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � 2�K�1/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

1fjPdjD1 �j.xj�2�K�1/j�2�KC3g;

becauseˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � 2�K�1/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.uj � 2�K�1/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� d � 2�KC2

and soˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � 2�K�1/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.uj � 2�K�1/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� 1

2

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � 2�K�1/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

:

Hence,

X

�d

2KdZ

Rdn27I

ˇ

ˇ

ˇ

ˇ

ˇ

Z

I

d�1Y

iD1.xi � ui/

�1!

.xd � ud/˛�1 (5.3.30)

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SknS�0 /\S�;1.x � u/ du

ˇ

ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd�Kkp�K˛p2�K.�.k�ı/p=.k�1/C1/.k�1/2�K.d�k/2�K..�˛�ı/pC1/

� Cp;

whenever ı D .k C ˛ � k˛/=k, p > k=.k C ˛/, thus p > .d � 1/=.d � 1C ˛/. (Letı D 1 if k D 1.)

Similarly,

X

�d

2Kd=pZ

I

d�1Y

iD1.xi � ui/

�1!

.xd � ud/˛�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SknS�0 /nS�;1.x � u/ du

�X

�d

2Kd=pZ

I.xi � ui/

�1�˛

d�1Y

iD2.xi � ui/

�1!

.xd � ud/˛�11Sk.x � u/ du

� CX

�d

2Kd=pZ

I

kY

iD1.xi � ui/

�1�˛=k

!

dY

iDkC1.xi � ui/

˛=.d�k/�1!

1Sk.x � u/ du

� CX

�d

2Kd=p�Kk�K˛

kY

iD1.xi � 2�K�1/�.kC˛/=k1fxi�2�KC1g

!

dY

iDkC11fxi<2�KC3g

!

and

X

�d

2KdZ

Rdn27I

ˇ

ˇ

ˇ

ˇ

ˇ

Z

I

d�1Y

iD1.xi � ui/

�1!

.xd � ud/˛�1 (5.3.31)

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SknS�0 /nS�;1 .x � u/ du

ˇ

ˇ

ˇ

ˇ

ˇ

p

dx

� Cp2Kd�Kkp�K˛p2�K.�.kC˛/p=kC1/k2�K.d�k/

� Cp;

if p > .d � 1/=.d � 1C ˛/. This yields that

d�1X

kD2

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1SknS�0 .x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp:

Step 4. The inequality

Z

Rdn27IsupT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1S1nS0.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp

can be proved in the same way.


If k D d and T � 2K , then instead of Lemma 5.2.15 we use Lemma 5.2.13 toobtain

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1SdnS0.x � u/ du

ˇ

ˇ

ˇ

ˇ

(5.3.32)

� CX

�d

2Kd=p�K˛Z

I

dY

iD1.xi � ui/

�1!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SdnS0/\S�;1.x � u/C 1.SdnS0/nS�;1 .x � u/�

du:

Then

X

�d

2Kd=p�K˛Z

I

dY

iD1.xi � ui/

�1!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SdnS0/\S�;1.x � u/ du

� CX

�d

2Kd=p�K˛Z

I

dY

iD2.xi � ui/

�1C.ı�1/=.d�1/!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

1SdnS0.x � u/ du:

On the other hand,

X

�d

2Kd=p�K˛Z

I

dY

iD1.xi � ui/

�1!

(5.3.33)

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SdnS0/nS�;1.x � u/ du

�X

�d

2Kd=p�K˛Z

I

dY

iD1.xi � ui/

�1�˛=d

!

1Sd.x � u/ du

and the inequality

Z

Rdn27Isup

T�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1SdnS0.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx � Cp

can be seen as in (5.3.30) and (5.3.31) for p > d=.d C ˛/.Now suppose that T < 2K . In this case we will use Lemma 5.2.16. As in (5.3.17),

we get by integrations by parts that

Z

Ia.u/K1;�

T;�0 .x � u/1SdnS0.x � u/ du

DdX

lD1

Z

Il

: : :

Z

Id

Al.u.l//.@lK

1;�T;�0 1SdnS0/.x � u.l// dul : : : dud;

where u.l/ WD .�; : : : ; �; ul; : : : ; ud/. Remark that Ad.�; : : : ; �/ D R

I a D 0. ByLemma 5.2.16,

supT<2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1SdnS0.x � u/ du

ˇ

ˇ

ˇ

ˇ

� CX

�d

dX

lD12Kd=p�Kl�K˛CK

Z

Il

: : :

Z

Id

dY

iD1.xi � u.l/i /

�1!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � u.l/j /

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛

1.SdnS0/\S�;1.x � u.l//C 1.SdnS0/nS�;1.x � u.l//�

dul : : : dud C CdX

lD12Kd=p�Kl

Z

Il

: : :

Z

Id

dY

iD1.xi � u.l/i /

�1!

xd � u.l/d

��11.SdnS0/\S�0 ;d.x � u.l// dul : : : dud:

The first sum can be handled as (5.3.32). Since u 2 I, x � u 2 Sd \S�0 ;d implies thatxi � 2�KC1 .i D 1; : : : ; d/ and x1 is in an interval I0

1 with length 4xd C C2�K�1 �Cxd. We estimate the second sum by

CdX

lD12Kd=p�Kl

Z

Il

: : :

Z

Id

dY

iD1.xi � 2�K�1/�1

!

�

xd � 2�K�1��1 1Sd\S�0 ;d.x � u.l// dul : : : dud

� C2Kd=p�Kd�K

d�1Y

iD2.xi � 2�K�1/�1�1=d1fxi�2�KC1g

!

�

xd � 2�K�1��2�2=d1fxd�2�KC1g1I0

1.x1/; (5.3.34)

consequently, integrating first in x1,Z

Rdn27Isup

T<2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1SdnS0.x � u/ du

ˇ

ˇ

ˇ

ˇ

p

dx

� Cp C Cp

Z

Rd�1

2Kd�Kdp�Kp

d�1Y

iD2.xi � 2�K�1/�.dC1/p=d1fxi�2�KC1g

!

.xd � 2�K�1/�.2dC2/p=dC11fxd�2�KC1g dx2 : : : dxd

� Cp C Cp2Kd�Kdp�Kp2�K.�.dC1/p=dC1/.d�2/2�K.�.2dC2/p=dC2/

� Cp;

whenever p > ddC1 . This finishes the proof of (5.3.2). �

Proof of (5.3.3) for q D 1 and d � 3 For the weak type inequality (5.3.3), we haveto show that

sup�>0

�d=.dC˛/�.�1;�� a > �;Rd n 27I/ � C

for all d=.d C ˛/-atoms a and ˛ � 1. Observe that

�d=.dC˛/�.jgj > �/ �Z

jgjd=.dC˛/

implies that we have to show only that

�d=.dC˛/��

supT>0

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T .x � u/1SdnS0.x � u/ du

ˇ

ˇ

ˇ

ˇ

> �;Rd n 27I�

� C:

Step 5. Similar to (5.3.33) with p D d=.d C ˛/,

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1.SdnS0/nS�;1.x � u/ du

ˇ

ˇ

ˇ

ˇ

� CX

�d

2KdZ

I.x1 � u1/

�1�˛

dY

iD2.xi � ui/

�1!

1Sd.x � u/ du

� C.x1 � 2�K�1/�1�˛1fx1�2�KC1g

dY

iD2.xi � 2�K�1/�11fxi�2�KC1g

!

:

If this is greater than C�, then by translation, we may suppose that

xd � ��1x�1�˛1

d�1Y

iD2x�1

i

!

(5.3.35)

and each xi is positive. We assume that

0 � xd < : : : < xkC1 < ��1=.dC˛/ < xk < : : : < x1 (5.3.36)

for some k D 0; 1 : : : ; d. The case k D d contradicts (5.3.35). For another k and forsome 0 � � � 1,

xd D x�dx1��d �

d�1Y

iDkC1x�=.d�k�1/

i

!

��1x�1�˛1

d�1Y

iD2x�1

i

!1��:

ThenZ

1fx�1�˛1

QdiD2 x�1

i ��g dx

�Z

��1

d�1Y

iDkC1x�=.d�k�1/C��1

i

!

kY

iD1x�1�˛=k

i

!1��dx1 : : : dxd�1

� ��1�� 1dC˛ .

�d�1�k C�/.d�1�k/�� 1

dC˛ .� kC˛k .1��/C1/k

D �� ddC˛ ;

whenever we choose � such that � kC˛k .1� �/C 1 < 0. If k D 0, then let � D 1 and

if k D d � 1, then � D 0.On the other hand,

supT�2K

ˇ

ˇ

ˇ

ˇ

Z

Ia.u/K1;�

T;�0 .x � u/1.SdnS0/\S�;1.x � u/ du

ˇ

ˇ

ˇ

ˇ

� CX

�d

2KdZ

I.xi � ui/

�1Cı

dY

iD2.xi � ui/

�1!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � uj/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

1SdnS0.x � u/ du

� CX

�d

.x1 � 2�K�1/�1Cı1fx1�2�KC1g

dY

iD2.xi � 2�K�1/�11fxi�2�KC1g

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�j.xj � 2�K�1/

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

1fjPdjD1 �j.xj�2�K�1/j�˛�ı�2�KC3g:

We may suppose again that

x�1Cı1

dY

iD2x�1

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

� �

and that (5.3.36) holds. Then

kY

iD2x�1C.ı�1/=.k�1/

i

!

dY

iDkC1x�1

i

!

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dX

jD1�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�˛�ı

� �:

By a transformation,

Z

1nx�1Cı1 .

QdiD2 x�1

i /jPdjD1 �jxjj�˛�ı��

o dx

�Z

1n.Qk

iD2 t�1C.ı�1/=.k�1/i /.

QdiDkC1 t�1i /jt1j�˛�ı��

o dt:

Assume that ��1=.dC˛/ < jt1j. The case k D d is again impossible. In other cases,

Z

1n.Qk

iD2 t�1C.ı�1/=.k�1/i /.


o dt

�Z

��1

d�1Y

iDkC1t�=.d�k�1/C��1i

!

kY

iD2t�1C.ı�1/=.k�1/i

!1��jt1j.�˛�ı/.1��/ dt1 : : : dtd�1

� ��1�� 1dC˛ .

�d�1�k C�/.d�1�k/�� 1

dC˛ ..� k�ık�1 .1��/C1/.k�1/C.�˛�ı/.1��/C1/

D �� ddC˛

if we choose � and ı such that � k�ık�1 .1 � �/C 1 < 0 and .�˛ � ı/.1 � �/C 1 < 0.

The cases k D 0 (� D 1), k D 1 (ı D 1) and k D d � 1 (� D 0) are included again.

If xd < jt1j � ��1=.dC˛/, then k < d and

Z

1n.Qk

iD2 t�1C.ı�1/=.k�1/i /.


o dt

�Z

��1

d�1Y

iDkC1t�=.d�k/C��1i

!

jt1j�=.d�k/�.˛Cı/.1��/

kY

iD2t�1C.ı�1/=.k�1/i

!1��dt1 : : : dtd�1

� ��1�� 1dC˛ ..

�d�k C�/.d�1�k/C �

d�k �.˛Cı/.1��/C1/�� 1dC˛ .� k�ı

k�1 .1��/C1/.k�1/

D �� ddC˛ ;

assuming that �d�k � .˛ C ı/.1 � �/C 1 > 0 and � k�ı

k�1 .1 � �/C 1 < 0.If jt1j < xd and jt1j � ��1=.dC˛/, then

Z

1n.Qk

iD2 t�1C.ı�1/=.k�1/i /.


o dt

�Z

�� 1��˛Cı

dY

iDkC1t

�d�k � 1��

˛Cı

i

!

kY

iD2t�1C ı�1

k�1

i

!1��˛Cı

dt2 : : : dtd

� �� 1��˛Cı �� 1

dC˛ .�

d�k � 1��˛Cı C1/.d�k/�� 1

dC˛ .� k�ık�1

1��˛CıC1/.k�1/

D �� ddC˛

as �d�k � 1��

˛Cı C 1 > 0 and � k�ık�1

1��˛Cı C 1 < 0.

Finally, if T < 2K , then we have to investigate only (5.3.34) if ˛ D 1. In case

d�1Y

iD1x�1

i

!

x�2d 1I0

1.x1/ � �;

we have

xd � ��1=21I0

1.x1/

kY

iD2x�1�1.k�1/

i

!1=2 d�1Y

iDkC1x�1

i

!1=2

:

ThenZ

1�.Qd�1

iD1 x�1i /x�2

d 1I01.x1/��

� dx �Z

xd1f.Qd�1iD1 x�1

i /x�2d ��g dx2 : : : dxd

�Z

��1

d�1Y

iDkC1x�=.d�k�1/C�=2�1=2

i

!2

kY

iD2x�1=2�1=2.k�1/

i

!2.1��/dx2 : : : dxd�1

� ��1�� 1dC1 .

2�d�1�k C�/.d�1�k/C.� k

k�1 .1��/C1/.k�1/

D �� ddC1 ;

whenever we choose � such that � kC˛k .1 � �/ C 1 < 0. This finishes the proof

of (5.3.3). �

5.3.2 Proof of Theorem 5.3.3

Proof of Theorem 5.3.3 Let a be an arbitrary ball p-atom with support B D B.0; /and 2K�1 < � 2K .K 2 Z/. Obviously,

Z

Rdn4B

ˇ

ˇ�2;�� a.x/ˇ

ˇ

pdx

�1X

iD2

Z

B.0;.iC1/2K/nB.0;i2K /

supT�2�K

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ

pdx

C1X

iD2

Z

B.0;.iC1/2K/nB.0;i2K /

supT<2�K

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ

pdx

DW .A/C .B/;

where d=ˇ < p � 1. We use Taylor’s formula for g.t/ D b�0.T.x � t//:

g.t/ DN�1X

kD0

X

kik1Dk

@i11 : : : @

idd g.0/

dY

jD1

tijj

ijŠC

X

kik1DN

@i11 : : : @

idd g.�t/

dY

jD1

tijj

ijŠ

for some 0 < � < 1. Here

@i11 : : : @

idd g.t/ D .�1/kik1Tkik1@i1

1 : : : @iddb�0.T.x � t//:

By the definition of the atom,

�2;�T a.x/

D Td

.2�/d=2

Z

Ba.t/b�0.T.x � t// dt

D Td

.2�/d=2

Z

Ba.t/

0

@b�0.T.x � t// �N�1X

kD0

X

kik1Dk

@i11 : : : @

idd g.0/

dY

jD1

tijj

ijŠ

1

A dt

D Td

.2�/d=2

Z

Ba.t/

X

kik1DN

.�1/kik1Tkik1@i11 : : : @

iddb�0.T.x � �t//

dY

jD1

tijj

ijŠdt:

Recall that we can suppose that N � N. p/. Taking into account (5.3.4), we conclude

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ � CTNCdZ

Bja.t/j kT.x � �t/k�ˇ

2 ktkN2 dt:

If t 2 B and x 2 B.0; .i C 1/2K/ n B.0; i2K/ for some i � 2, then

kx � �tk2 � kxk2 � ktk2 � i2K � 2K D .i � 1/2K;

which implies

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ � CTNCd�ˇ.i � 1/�ˇ2K.N�ˇ/Z

Ija.t/j dt

� CTNCd�ˇ.i � 1/�ˇ2K.N�ˇ/2�Kd=pCKd

� C.i � 1/�ˇ2�Kd=p

if T � 2�K . Then

.A/ D1X

iD2

Z

B.0;.iC1/2K/nB.0;i2K /

supT�2�K

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ

pdx

� Cp

1X

iD2

Z

B.0;.iC1/2K/nB.0;i2K /

.i � 1/�ˇp2�Kd dx

D Cp

1X

iD2.i � 1/d�1�ˇp < 1;

as p > 1=ˇ. Similarly,

�2;�T a.x/

D Td

.2�/d=2

Z

Ba.t/

X

kik1DNC1.�1/kik1Tkik1@i1

1 : : : @iddb�0.T.x � �t//

dY

jD1

tijj

ijŠdt

and

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ � CTNC1CdZ

Bja.t/j kT.x � �t/k�ˇ

2 ktkNC12 dt

� CTNC1Cd�ˇ.i � 1/�ˇ2K.NC1�ˇ/Z

Ija.t/j dt

� C.i � 1/�ˇ2�Kd=p

if T < 2�K . The inequality B < 1 can be shown as above.For p D d=ˇ let

E� WD ˚

i � 2 W .i � 1/�ˇ > �C�12Kˇ�

and observe that

�d=ˇ �

supT�2�K

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ > �;R n 4B

!

� C�d=ˇX

i2E�

.i � 1/d�12Kd:

If k is the largest integer, for which .i � 1/�ˇ > �C�12Kˇ, then

�d=ˇ �

supT�2�K

ˇ

ˇ

ˇ�2;�T a.x/

ˇ

ˇ

ˇ > �;R n 4I

!

� C�d=ˇkd2Kd � C:

The term supT<2�K j�2;�T a.x/j can be estimated similarly. Now our result followsfrom Theorems 3.6.1 and 3.6.4. �

5.3.3 Some Summability Methods

As we mentioned above, Examples 2.11.3–2.11.11 all satisfy the condi-tions (5.1.1), (5.1.7) and (5.1.8). Thus Theorem 5.3.2 and its consequences aretrue for those examples and for q D 1 and q D 1.

For q D 2 we investigated Examples 5.2.17, 5.2.18 and 5.2.19. These examplessatisfy (5.1.1) and (5.1.6) if ˛ > .d �1/=2. We have to check here (5.3.4) or (5.3.6).The following result can be verified by an easy calculation.

Theorem 5.3.8 The examples �0.t/ D e�ktk22=2, �0.t/ D e�ktk2 and �0.t/ D .1 Cktk22/�.dC1/=2 satisfy (5.3.6) for all N 2 N. Thus, for all 0 1I0; if ktk2 � 1

with D 2, b�0.x/ as well as all of its derivatives can be estimated by kxk�d=2�˛�1=22 .

Corollary 5.3.9 For the Bochner-Riesz summation and for all i1; : : : ; id � 0, wehave

ˇ

ˇ

ˇ@i11 � � � @id

db�0.x/

ˇ

ˇ

ˇ � C kxk�d=2�˛�1=22 .x ¤ 0/:

Note the same result holds for 2 P (see Lu [233, p. 132]).

Theorem 5.3.10 Consider q D 2 and the Riesz summation with 2 P and ˛ >.d � 1/=2. If d=.d=2C ˛ C 1=2/ 0

��.�q;�� f > �/.d=2C˛C1=2/=d � C k f kH�

d=.d=2C˛C1=2/:

Proof Let us choose N 2 N such that

N < ˛ � .d � 1/=2 � N C 1:

Then (5.3.4) holds with ˇ D d=2C ˛ C 1=2. �Of course Corollaries 5.3.6 and 5.3.7 hold as well for all summability methods

just considered.

5.3.4 Further Results for the Bochner-Riesz Means

The boundedness of the operator �2;˛� is complicated and not completely solved ifq D D 2 and ˛ � .d � 1/=2. Here we summarize the corresponding results. Agood overview and the proofs of this topic can be found in the books of Grafakos[152, 154, 155], Lu and Yan [239] and Stein and Weiss [311]. The following theoremis due to Tao [320].

Theorem 5.3.11 If d � 2, 0 < ˛ � .d � 1/=2 and

1 

2d

d � 1 � 2˛;

then the Bochner-Riesz maximal operator �2;˛� is not bounded from Lp.Rd/ to

Lp;1.Rd/ (see Fig. 5.15).By Theorems 5.3.11 and 4.2.9, Fig. 5.15 shows the region where �2;˛�

is unbounded from Lp.Rd/ to Lp;1.Rd/. Obviously, the operator �2;˛� is

unbounded from Lp.Rd/ to Lp.R

d/ on the same region. Note that the exactregion of the boundedness or unboundedness of �2;˛� is still unknown (seeFig. 5.18).

Stein [311, p. 276] proved that Theorem 5.2.24 holds also for the maximaloperator �2;˛� .

Fig. 5.15 Unboundedness of �2;˛�

from Lp.Rd/ to Lp;1.Rd/

Theorem 5.3.12 If d � 2, 0 < ˛ � .d � 1/=2 and

2.d � 1/d � 1C 2˛

< p <2.d � 1/

d � 1 � 2˛ ;

then the Bochner-Riesz maximal operator �2;˛� is bounded on Lp.Rd/ (see Fig. 5.11).

Carbery improved this theorem in [48] for d D 2.

Theorem 5.3.13 If d D 2, 0 < ˛ � 1=2 and

2

1C 2˛< p <

4

1 � 2˛;


Christ [61] generalized this result to higher dimensions.


d � 1

2.d C 1/� ˛ � d � 1

2and

2.d � 1/

d � 1C 2˛< p <

2d

d � 1 � 2˛ ;


The following result follows from analytic interpolation and from Theo-rems 5.3.12 and 5.3.14 (see e.g. Stein and Weiss [311, p. 276, p. 205]).


0 < ˛ <d � 1

2.d C 1/and

2.d � 1/

d � 1C 2˛< p <

2.d � 1/

d � 1 � 4˛ ;

Fig. 5.16 Boundedness of�2;˛�

from Lp.R2/ to

Lp;1.R2/ when d D 2


then the Bochner-Riesz maximal operator �2;˛� is bounded on Lp.R

d/ (seeFig. 5.17).

It is still an open question as to whether �2;˛� is bounded or unbounded in theregion of Fig. 5.18. If d D 2, then the question is open on the right-hand side of theregion of Fig. 5.18 only, i.e. for 1=p � 1=2.

Fig. 5.17 Boundedness of �2;˛�

from Lp.Rd/ to Lp;1.Rd/ when d � 3

Fig. 5.18 Open question of the boundedness of �2;˛�

when d � 3

Of course, in Theorems 5.3.12–5.3.15, �2;˛� is also bounded from Lp.Rd/ to

Lp;1.Rd/. This implies that

limT!1 �

2;˛T f D f a.e.

Figures 5.16 and 5.17 show the region where �2;˛� is bounded from Lp.Rd/ to

Lp;1.Rd/ and almost everywhere convergence hold.By Theorem 1.2.6, if �2;˛� is of weak type . p; p/, i.e. it is bounded from Lp.R

d/

to Lp;1.Rd/, then limT!1 �2;˛T f D f almost everywhere for all f 2 Lp.R

d/. Theconverse is also true when 1 � p � 2: almost everywhere convergence impliesthat the corresponding maximal operator is of weak type . p; p/. More exactly, ifX D Lp.R

d/, 1 � p � 2 and Tn commutes with translation, then the converse ofTheorem 1.2.6 holds (see Stein [307]). However, this result is no longer true forp > 2. The preceding theorems concerning the almost everywhere convergencewere generalized by Carbery et al. [50] (see Fig. 5.19).

Theorem 5.3.16 If d � 2, 0 < ˛ � .d � 1/=2 and

2.d � 1/d � 1C 2˛

< p <2d

d � 1 � 2˛ ;

then for all f 2 Lp.Rd/

limT!1 �

2;˛T f D f a.e.

The theorem is not true for p D 1 and ˛ D .d � 1/=2. Stein [307] (see alsoGrafakos [152, 154, 155], Lu and Yan [239] and Stein and Weiss [311]) found an

Fig. 5.19 Almosteverywhere convergence of�2;˛T f , f 2 Lp.R

d/

integrable function f 2 L1.Rd/ such that

lim supT!1

ˇ

ˇ

ˇ�2;.d�1/=2T f

ˇ

ˇ

ˇ D 1 a.e.

The convergence of the preceding theorem holds also for some special functionsf 2 L log L.Rd/. For the exact result see Chen and Fan [59].

5.4 Convergence at Lebesgue Points

Here we generalize the results of Sect. 2.9. First we generalize the concept ofLebesgue points. We introduce multi-dimensional Lebesgue points for the circularsummability, the modified and modified strong Lebesgue points for the cubic andtriangular summability. As main result, it is proved that the �-means of a functionf 2 W.L1; `1/.Rd/ convergence to f at each (modified strong) Lebesgue point.Concerning to this section we refer to Butzer and Nessel [46], Stein and Weisz[311] and Weisz [113, 374–376].

5.4.1 Circular Summability .q D 2/

The results for the circular summability are similar to the one-dimensional onespresented in Sects. 2.7 and 2.9, so the proofs are left to the reader.

Recall that in Sect. 3.1, the Hardy-Littlewood maximal function was defined by

Mpf .x/ WD suph>0

�

1

.2h/d

Z h

�h� � �Z h

�hj f .x � s/jp ds

�1=p

:

Moreover, for all locally integrable functions f ,

limh!0

1

.2h/d

Z h

�h� � �Z h

�hf .x � s/ ds D f .x/ a.e. x 2 R

d:

Definition 5.4.1 A point x 2 Rd is called a p-Lebesgue point of f if

limh!0

�

1

.2h/d

Z h

�h� � �Z h

�hj f .x � s/ � f .x/jp ds

�1=p

D 0 .1 � p < 1/ :

Again, p < r implies that all r-Lebesgue points are p-Lebesgue points.

Theorem 5.4.2 Almost every point x 2 Rd is a p-Lebesgue point of f 2

W.Lp; `1/.Rd/ .1 � p < 1/.

Herz spaces are introduced as in the one-dimensional case.

Definition 5.4.3 A function f 2 Llocq .R

d/ is in the Herz space Eq.Rd/, resp. in the

homogeneous Herz space PEq.Rd/, if

k f kEqWD

f1B.0;1/

qC

1X

kD12kd.1�1=q/ k f1Pkkq < 1;

resp.,

k f k PEqWD

1X

kD�12kd.1�1=q/ k f1Pk kq < 1;

where Pk WD B.0; 2k/ n B.0; 2k�1/, .k 2 Z/ and 1 � q � 1.Recall that B.0; c/ D ˚

x 2 Rd W kxk2 < c

�

. As for one dimension,

Eq.Rd/ � PEq.R

d/; Eq.Rd/ � Lq.R

d/; k f k PEq� Cq k f kEq

; k f kq � Cq k f kEq;

L1.Rd/ D PE1.Rd/ PEq.R

d/ PEq0.Rd/ PE1.Rd/ 1 < q < q0 < 1

and

Eq.Rd/ D Lq.R

d/\ PEq.Rd/ and k f kEq � k f kq C k f k PEq

with equivalent norms.

Theorem 5.4.4 Let �.x/ WD supktk2�kxk2 j f .t/j. Then f 2 PE1.Rd/ if and only if� 2 L1.Rd/ and

C�1k�k1 � k f k PE1

� Ck�k1:

Theorem 5.4.5 Let � 2 L1.Rd/, 1 � p < 1 and 1=p C 1=q D 1. Ifb�0 2 PEq.Rd/,

then for all f 2 Llocp .R

d/,

�2;�� f .x/ � C

b�0

PEq

Mpf .x/:

Note that �0 was introduced in (5.1.2).

Theorem 5.4.6 Let � 2 L1.Rd/, 1 � p � 1 and 1=p C 1=q D 1. Ifb�0 2 PEq.Rd/,

then

�2;�� f

W.Lp;1 ;`1/� Cp

b�0

PEq

k f kW.Lp ;`1/

for all f 2 W.Lp; `1/.Rd/. Moreover, for every p < r � 1,

�2;�� f

W.Lr ;`1/� Cr

b�0

PEq

k f kW.Lr ;`1/ . f 2 W.Lr; `1/.Rd//:

The result about the convergence at Lebesgue points reads as follows.

Theorem 5.4.7 Let � 2 L1.Rd/, 1 � p < 1 and 1=p C 1=q D 1. Ifb�0 2 PEq.Rd/,

then

limT!1 �

2;�T f .x/ D f .x/

for all p-Lebesgue points of f 2 W.Lp; `1/.Rd/.As in the one-dimensional case (see Theorem 2.9.5), the converse holds also.

Theorem 5.4.8 Suppose that � 2 L1.Rd/, b�0 2 L1.Rd/, 1 � p < 1 and 1=p C1=q D 1. If

limT!1 �

2;�T f .x/ D f .x/

for all p-Lebesgue points of f 2 Lp.Rd/, thenb�0 2 PEq.R

d/.

Note that the examples in Sect. 5.2.2 satisfy the conditionb�0 2 PEq.Rd/ for all

1 � q � 1.

5.4.2 Cubic and Triangular Summability (q D 1 and q D 1)

Here we will use the conditions from Sect. 5.1, more exactly, we assume theconditions (5.1.7) and (5.1.8). The modified maximal function Mpf was introducedin Sect. 3.1 as

Mpf .x/

WD supP2i1 h;:::;2id h

;i2Nd ;h>0

2��kik1

1ˇ

ˇP2i1h;:::;2id h

ˇ

ˇ

Z

P2i1 h;:::;2id h

j f .x � s/jp ds

!1=p

;

where the supremum is taken over all parallelepipeds P2i1h;:::;2id h .i 2 Nd; h > 0/

whose centre is the origin and whose sides are parallel to the axes and/or tothe diagonals. If all parallelepipeds have sides parallel to the axes, we obtain thedefinition of M.1/

p f :

M.1/p f .x/ WD sup

i2Nd ;h>0

2��kik1

1

.2h/d2kik1

Z 2i1h

�2i1h� � �Z 2id h

�2id hj f .x � s/jp ds

!1=p

:

In the two-dimensional case Mpf is basically the sum of M.1/p f and of

M.2/p f .x; y/

WD supi;j2N;h>0

2��.iCj/

1

4 � 2iCjh2

Z 2ih

�2ih

Z sC2jh

s�2jhj f .x � s; y � t/jp dt ds

!1=p

:

Starting from the maximal function Mpf , we introduce

Ur;pf .x/ WD U�r;p f .x/ WD sup

P2i1 h;:::;2id h

;i2Nd ;h>0

2ik h<r;kD1;:::d

2��kik1

1ˇ

ˇP2i1h;:::;2id h

ˇ

ˇ

!1=p

Z

P2i1 h;:::;2id h

j f .x � s/� f .x/jp ds

!1=p

;

where the supremum is taken over all parallelepipeds whose centre is the origin andwhose sides are parallel to the axes and/or to the diagonals. Obviously,

Ur;pf .x/ � supi2Nd ;h>0;2ik h<r;kD1;:::d

2��kik1�

1

.2h/d2kik1

�1=p

Z 2i1h

�2i1h

Z ı1s1C2i2h

ı1s1�2i2h� � �Z ıd�1.s1�s2��sd�1/C2id h

ıd�1.s1�s2��sd�1/�2id hj f .x � s/� f .x/jp ds

!1=p

;

where ıi D 0; 1 .i D 1; : : : ; d � 1/. Taking the supremum in the definition ofUr;pf over all parallelepipeds whose sides are parallel to the axes, we obtain thedefinition

U.1/r;p f .x/ WD sup

i2Nd ;h>0;2ik h<r;kD1;:::d2��kik1

�

1

.2h/d2kik1

�1=p

Z 2i1h

�2i1h� � �Z 2id h

�2id hj f .x � s/ � f .x/jp ds

!1=p

:

In case p D 1 we omit the notation p and write simply Urf and U.1/r f .

In the two-dimensional case Ur;pf .x; y/ is basically the sum of U.1/r;p f .x; y/

and of

U.2/r;p f .x; y/

WD supi;j2N;h>02ih<r;2jh<r

2��.iCj/

�

1

4 � 2iCjh2

�1=p

Z 2ih

�2ih

Z sC2jh

s�2jhj f .x � s; y � t/ � f .x; y/jp dt ds

!1=p

:

Note that the definition of the p-Lebesgue points (see Definition 5.4.1) can berewritten as

limr!0

sup0<h<r

�

1

.2h/d

Z h

�h� � �Z h

�hj f .x C s/� f .x/jp ds

�1=p

D 0:

Using this definition, we introduce other type of Lebesgue points.

Definition 5.4.9 We say that a point x 2 Rd is a modified p-Lebesgue point of

f 2 Llocp .R

d/ .1 � p < 1/ if for all � > 0

limr!0

U.1/r;p f .x/ D 0:

If in addition,

limr!0

Ur;pf .x/ D 0;

then we say that x 2 Rd is a modified strong p-Lebesgue point. If p D 1, then we

call the points modified Lebesgue points or modified strong Lebesgue points.It is equivalent if we suppose that the corresponding limits are equal to 0 for all

small numbers � > 0. Indeed, limr!0 U�2r;p f .x/ D 0 implies limr!0 U�1

r;p f .x/ D 0

for all �1 > �2 > 0. The same holds for U.1/r;p f . Obviously, every modified (strong)

p-Lebesgue point is a modified (strong) Lebesgue point. If f is continuous at x, thenx is a modified (strong) p-Lebesgue point of f for all 1 � p < 1.

Theorem 5.4.10 Almost every point x 2 Rd is a modified p-Lebesgue point and a

modified strong p-Lebesgue point of f 2 W.Lp; `1/.Rd/ .1 � p < 1/.

Proof It is enough to prove the theorem for the modified strong Lebesgue pointsand for f 2 Lp.R

d/. Let � > 0 be arbitrary. If f is a continuous function, then x isobviously a strong p-Lebesgue point. By Theorem 3.1.7,

�p�

�

supr>0

Ur;pf > �

�

� �p�.Mpf > �=2/C 2�p�.j f j > �=2/

� C k f kpp :

Since the result holds for continuous functions and the continuous functionsare dense in Lp.R

d/, the theorem follows from the usual density argument ofTheorem 1.2.6. �

It is not sure that a point .x; : : : ; x/ of the diagonal is a modified (strong) p-Lebesgue point of a general function f 2 W.Lp; `1/.Rd/ for almost every x 2 R.For the strong summability results of the next section, we need to investigate thisquestion and functions of type

f .x/ DdY

jD1f0.xj/

with a one-dimensional function f0.

Theorem 5.4.11 Suppose that f .x/ D QdjD1 f0.xj/. If xj . j D 1; : : : ; d/ is a p-

Lebesgue point of f0 2 W.Lp; `1/.R/, then x is a modified p-Lebesgue point off 2 W.Lp; `1/.Rd/ .1 � p < 1/.

Proof It is enough to show the theorem for d D 2. We have

1

4 � 2iCjh2

Z 2ih

�2ih

Z 2jh

�2jhj f .x � s; y � t/ � f .x; y/jp dt ds

!1=p

�

1

4 � 2iCjh2

Z 2ih

�2ih

Z 2jh

�2jhj f0.x � s/ � f0.x/jp j f0.y � t/jp dt ds

!1=p

C

1

4 � 2iCjh2

Z 2ih

�2ih

Z 2jh

�2jhj f0.x/jp j f0.y � t/ � f0.y/jp dt ds

!1=p

D A1.x; y/C A2.x; y/:

It is easy to see that if y is a p-Lebesgue point of f0, then Mpf0.y/ is finite. Since x isalso a p-Lebesgue point of f0,

A1.x; y/ � Mpf0.y/

1

2 � 2ih

Z 2ih

�2ihj f0.x � s/ � f0.x/jp ds

!1=p

< �;

whenever 2ih < r and r is small enough. The term A2 can be handled similarly,

A2.x; y/ � C

1

2 � 2jh

Z 2jh

�2jhj f0.y � t/ � f0.y/jp dt

!1=p

< �;

whenever 2jh < r and r is small enough. �The following corollary can be seen in the same way.

Corollary 5.4.12 Suppose that f .x/ D QdjD1 f0.xj/. If xj . j D 1; : : : ; d/ is a p-

Lebesgue point of f0 2 W.Lp; `1/.R/, then M.1/p f .x/ is finite .1 � p < 1/.

Proof It is easy to see that M.1/p f .x/ � Qd

jD1 Mpf0.xj/. �For the modified strong Lebesgue points we need in addition that f0 is almost

everywhere locally bounded. Recall that f0 is locally bounded at x if there exists aneighborhood of x such that f0 is bounded on this neighborhood.

Theorem 5.4.13 Suppose that f .x/ D QdjD1 f0.xj/. If xj . j D 1; : : : ; d/ is a p-

Lebesgue point of f0 2 W.Lp; `1/.R/ and f0 is locally bounded at xj, then x is amodified strong p-Lebesgue point of f 2 W.Lp; `1/.Rd/ .1 � p < 1/.

Proof We will prove the theorem for d D 2 and for U.2/r;p f .x; y/, only. Obviously,

1

4 � 2iCjh2

Z 2ih

�2ih

Z sC2jh

s�2jhj f .x � s; y � t/ � f .x; y/jp dt ds

!1=p

�

1

4 � 2iCjh2

Z 2ih

�2ih

Z sC2jh

s�2jhj f0.x � s/ � f0.x/jp j f0.y � t/jp dt ds

!1=p

C

1

4 � 2iCjh2

Z 2ih

�2ih

Z sC2jh

s�2jhj f0.x/jp j f0.y � t/ � f0.y/jp dt ds

!1=p

D A3.x; y/C A4.x; y/:

Since x is a p-Lebesgue point of f0 and f0 is bounded in a neighborhood of y,

A3.x; y/ � C

1

2 � 2ih

Z 2ih

�2ihj f0.x � s/ � f0.x/jp ds

!1=p

< �;

whenever 2ih < r, 2jh < r and r is small enough.On the other hand,

A4.x; y/

D

1

4 � 2iCjh2

Z 2ihC2jh

�2ih�2jh

Z 2ih^.tC2jh/

�2ih_.t�2jh/j f0.x/jp j f0.y � t/ � f0.y/jp ds dt

!1=p

:

If i � j, then

A4.x; y/ � C

1

4 � 2iCjh2

Z 2iC1h

�2iC1h

Z tC2jh

t�2jhj f0.y � t/ � f0.y/jp ds dt

!1=p

D C

1

2 � 2ih

Z 2iC1h

�2iC1hj f0.y � t/ � f0.y/jp dt

!1=p

< �

and if i < j, then

A4.x; y/ � C

1

4 � 2iCjh2

Z 2jC1h

�2jC1h

Z 2ih

�2ihj f0.y � t/ � f0.y/jp ds dt

!1=p

D C

1

2 � 2jh

Z 2jC1h

�2jC1hj f0.y � t/ � f0.y/jp dt

!1=p

< �;

whenever 2ih < r, 2jh < r and r is small enough. �Note that in Theorems 5.4.11 and 5.4.13, we do not need the term 2��kik1 .

Moreover, these theorems remain true if we write

dY

jD1

�

f0.xj � sj/ � f0.xj/�

(5.4.1)

instead of f .x � s/ � f .x/, where f0 is a one-dimensional function.

Theorem 5.4.14 If xj . j D 1; : : : ; d/ is a p-Lebesgue point of the function f0 2W.Lp; `1/.R/ .1 � p < 1/, then

limr!1 sup

i2Nd ;h>0;2ik h<r;kD1;:::d2��kik1

�

1

.2h/d2kik1

�1=p

0

@

Z 2i1h

�2i1h� � �Z 2id h

�2id h

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dY

jD1

�

f0.xj � sj/ � f0.xj/�

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p

ds

1

A

1=p

D 0:

If in addition f0 is locally bounded at xj . j D 1; : : : ; d/, then

limr!1 sup

P2i1 h;:::;2id h;i2Nd ;h>0

2ik h<r;kD1;:::d

2��kik1

1ˇ

ˇP2i1h;:::;2id h

ˇ

ˇ

!1=p

0

@

Z

P2i1 h;:::;2id h

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

dY

jD1

�

f0.xj � sj/� f0.xj/�

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

p

ds

1

A

1=p

D 0:

Now we are ready to show that the �-means �q;�T f converge to f at each modified

strong Lebesgue point for both q D 1 and q D 1.

Theorem 5.4.15 Suppose that q D 1 or q D 1 and (5.1.7) and (5.1.8) are satisfiedfor some 0 < ˛ < 1. If f 2 W.L1; `1/.Rd/, x is a modified strong Lebesgue point

of f and Mf .x/ is finite for a suitable small number � > 0, then

limT!1 �

q;�T f .x/ D f .x/:

Since by Theorems 3.1.7 and 5.4.10 almost every point is a modified strongLebesgue point and the modified maximal function Mf is almost everywhere finitefor f 2 W.L1; `1/.Rd/ and � > 0, Theorem 5.4.15 implies

Corollary 5.4.16 Suppose that q D 1 or q D 1 and (5.1.7) and (5.1.8) aresatisfied for some 0 < ˛ < 1. If f 2 W.L1; `1/.Rd/, then

limT!1 �

q;�T f .x/ D f .x/ a.e.

In the next theorem we do not need the modified maximal function Mf .

Theorem 5.4.17 Suppose that q D 1 and (5.1.7) and (5.1.8) are satisfied for some0 < ˛ < 1. If f 2 W.Lp; `1/.Rd/ .1 0, then

limT!1 �

q;�T f .x/ D f .x/:

In order to prove the strong summability results in the next section, we state twoadditional theorems for q D 1 and for the function (5.4.1). Taking into accountCorollary 5.4.12, Theorem 5.4.14 and the proof of Theorem 5.4.17, we can provethe next result exactly as Theorem 5.4.17. The details are left to the reader.

Theorem 5.4.18 Suppose that q D 1 and (5.1.7) and (5.1.8) are satisfied forsome 0 < ˛ < 1. If f0 2 W.Lp; `1/.R/ .1 < p < 1/ and xj . j D 1; : : : ; d/ is ap-Lebesgue point of f0, then

limT!1

Z

Rd

dY

jD1

�


K1;�T .s/ ds D 0:

If in addition f0 is locally bounded at xj, then the preceding theorem can beextended to p D 1.

Theorem 5.4.19 Suppose that q D 1 and (5.1.7) and (5.1.8) are satisfied for some0 < ˛ < 1. If f0 2 W.L1; `1/.R/, f0 is locally bounded at xj and xj . j D 1; : : : ; d/is a Lebesgue point of f0, then

limT!1

Z

Rd

dY

jD1

�


K1;�T .s/ ds D 0:

The proofs of Theorems 5.4.15, 5.4.17 and 5.4.19 will be given in the nextsubsections.

5.4.2.1 Proof of the Results for q D 1 and d D 2

We will use the inequalities from (5.1.4) and Lemma 5.2.11:

ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT2; (5.4.2)ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � Cs�1t�1; (5.4.3)ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT�˛s�1t�1.s � t/�˛; (5.4.4)ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT1�˛s�1.s � t/�˛: (5.4.5)

Proof of Theorem 5.4.15 for q D 1 and d D 2 Recall that by (5.1.3),

Kq;�T .s; t/ WD 1

2�T2b

�.q/0 .Ts;Tt/;

where q D 1;1. Since �.q/0 ;b

�.q/0 2 L1.Rd/ by Theorem 5.2.1, the Fourier inversion

formula yields that

Z

R2

Kq;�T .s; t/ ds dt D 1

2�

Z

R2

b

�.q/0 .s; t/ ds dt D �.0/ D 1:

Thusˇ

ˇ

ˇ�q;�T f .x; y/� f .x; y/

ˇ

ˇ

ˇ

�Z

R2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇKq;�T .s; t/

ˇ

ˇ

ˇ ds dt: (5.4.6)

It is enough to integrate over the set f.s; t/ 2 R2 W s > t > 0g. Let us decompose

this set into the union [5iD1Ai, where

A1 WD f.s; t/ W 0 < s � 2=T; 0 < t < sg;A2 WD f.s; t/ W s > 2=T; 0 < t � 1=Tg;A3 WD f.s; t/ W s > 2=T; 1=T < t � s=2g;A4 WD f.s; t/ W s > 2=T; s=2 < t � s � 1=Tg;A5 WD f.s; t/ W s > 2=T; s � 1=T < t � sg:

Fig. 5.20 The sets Ai

The sets Ai can be seen on Fig. 5.20. Let � < ˛=2 ^ 1. Since .x; y/ is a modifiedstrong Lebesgue point of f , we can fix a number r < 1 such that

Urf .x; y/ < �:

Let us denote the square Œ0; r=2� Œ0; r=2� by Sr=2 and let 2=T < r=2. We willintegrate the right-hand side of (5.4.6) over the sets

5[

iD1.Ai \ Sr=2/ and

5[

iD1.Ai \ Sc

r=2/:

Of course, A1 � Sr=2. By (5.4.2),

Z

A1

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� CT2Z 2=T

0

Z 2=T

0

j f .x � s; y � t/ � f .x; y/j ds dt

� CU.1/r f .x; y/ < C�:

Let us denote by r0 the largest number i, for which r=2 � 2iC1=T < r. By (5.4.5),

Z

A2\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1T1�˛

�

2i

T

��1 �2i

T� 1

T

��˛

Z 2iC1=T

2i=T

Z 1=T

0

j f .x � s; y � t/ � f .x; y/j ds dt

� Cr0X

iD12.��˛/i2�� i

�

T2

2i

�Z 2iC1=T

2i=T

Z 1=T

0

j f .x � s; y � t/ � f .x; y/j ds dt

� Cr0X

iD12.��˛/iU.1/

r f .x; y/ < C�;

because � < ˛.Since s � t � s=2 and s � t � t on A3, we obtain by (5.4.4) that

ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT�˛s�1�˛=2t�1�˛=2: (5.4.7)

HenceZ

A3\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1

i�1X

jD0T�˛

�

2i

T

��1�˛=2 �2j

T

��1�˛=2

Z 2iC1=T

2i=T

Z 2jC1=T

2j=Tj f .x � s; y � t/ � f .x; y/j ds dt

� Cr0X

iD1

i�1X

jD02.��˛=2/.iCj/

2��.iCj/

�

T2

2iCj

�Z 2iC1=T

2i=T

Z 2jC1=T


� Cr0X

iD1

i�1X

jD02.��˛=2/.iCj/U.1/

r f .x; y/ < C�:

Since t > s=2 on A4, (5.4.4) implies

ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT�˛s�1�ˇ .s � t/ˇ�˛�1 (5.4.8)

with 0 � ˇ � 1. Then

Z

A4\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1

i�1X

jD0T�˛

�

2i

T

��1�ˇ �2j

T

�ˇ�˛�1

Z 2iC1=T

2i=T

Z s�2j=T

s�2jC1=Tj f .x � s; y � t/ � f .x; y/j dt ds

� Cr0X

iD1

i�1X

jD02.��ˇ/i2.��˛Cˇ/j

2��.iCj/

�

T2

2iCj

�Z 2iC1=T

2i=T

Z s�2j=T


� Cr0X

iD1

i�1X

jD02.��ˇ/i2.��˛Cˇ/jU.2/

r f .x; y/ < C�;

where we choose ˇ D ˛=2 if 0 < ˛ � 2 and ˇ D 1 if ˛ > 2.We get from (5.4.3) that

ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � Cs�2

on the set A5. This implies

Z

A5\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1

�

2i

T

��2 Z 2iC1=T

2i=T

Z s

s�1=Tj f .x � s; y � t/ � f .x; y/j dt ds

� Cr0X

iD12.��1/i2�� i

�

T2

2i

�Z 2iC1=T

2i=T

Z s


� Cr0X

iD12.��1/iU.2/

r f .x; y/ < C�:

Similarly, we can show that

Z

A2\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

2.��˛/iM.1/f .x; y/C C1X

iDr0

2�˛if .x; y/

� C2.��˛/r0M.1/f .x; y/C C2�˛r0 f .x; y/

� C.Tr/��˛M.1/f .x; y/C C.Tr/�˛ f .x; y/ ! 0

andZ

A3\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

i�1X

jD02.��˛=2/.iCj/M.1/f .x; y/C C

1X

iDr0

i�1X

jD02�˛=2.iCj/f .x; y/

� C2.��˛=2/r0M.1/f .x; y/C C2�˛=2r0 f .x; y/ ! 0

andZ

A4\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

i�1X

jD02.��ˇ/i2.��˛Cˇ/jM.2/f .x; y/C C

1X

iDr0

i�1X

jD02�ˇi2.�˛Cˇ/jf .x; y/

� C2.��ˇ/r0M.2/f .x; y/C C2�ˇr0 f .x; y/ ! 0;

as T ! 1, where ˇ is chosen as before. Finally,

Z

A5\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

2.��1/iM.2/f .x; y/C C1X

iDr0

2�if .x; y/

� C2.��1/r0M.2/f .x; y/C C2�r0 f .x; y/ ! 0;

as T ! 1. Note that A1 \ Scr=2 D ;. �

Proof of Theorem 5.4.17 for d D 2 We have to integrate the integral in (5.4.6) againon the set [5

iD1Ai. Now let � < ˛=2 ^ 1=2 ^ 1=.2q/, where 1=p C 1=q D 1.

Since .x; y/ is a modified p-Lebesgue point of f , we can fix a number r such thatU.1/

r;p f .x; y/ < �. Since

U.1/r;1 f � U.1/

r;p f and M.1/f � M.1/p f ; (5.4.9)

we can prove in the same way as in Theorem 5.4.15 that

Z

Ai\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt < C�

andZ

Ai\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C2.��˛=2/r0M.1/f .x; y/C C2�˛=2r0 f .x; y/

� C.Tr/��˛=2M.1/f .x; y/C C.Tr/�˛=2f .x; y/ ! 0;

for i D 1; 2; 3, as T ! 1.So we have to consider the sets A4 and A5, only. It is easy to see that

Z

A4\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

�r0X

iD1

iX

jDi�1Z 2iC1=T

2i=T

Z 2jC1=T

2j=Tj f .x � s; y � t/ � f .x; y/j ˇˇK�

T .s; t/ˇ

ˇ 1A4.s; t/ dt ds:

By (5.4.8) and Hölder’s inequality,

Z

A4\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

�r0X

iD1

iX

jDi�1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=Tj f .x � s; y � t/ � f .x; y/jp dt ds

!1=p

Z 2iC1=T

2i=T

Z s�1=T

2i�1=TT�˛qs.�1�ˇ/q.s � t/.ˇ�˛�1/q1A4.s; t/ dt ds

!1=q

:

Choosing ˇ such that .ˇ � ˛ � 1/q C 1 < 0, i.e. ˇ < ˛ C 1 � 1=q, we conclude

Z 2iC1=T

2i=T

Z s�1=T

2i�1=TT�˛qs.�1�ˇ/q.s � t/.ˇ�˛�1/q dt ds

� CT�˛q

�

1

T

�.ˇ�˛�1/qC1 �2i

T

�.�1�ˇ/qC1

� C

�

T

2i

�2q�22�i.1�.1�ˇ/q/:

Furthermore, if we choose ˇ such that 1 � .1 � ˇ/q > 0, i.e. 1 � 1=q < ˇ, then

Z

A4\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cp

r0X

iD1

iX

jDi�12.��1=2C.1�ˇ/q=2/.iCj/

2��.iCj/

T2

2iCj

Z 2iC1=T

2i=T

Z 2jC1=T


!1=p

� Cp

r0X

iD1

iX

jDi�12.��1=2C.1�ˇ/q=2/.iCj/U.1/

r;p f .x; y/ < C�:

We want to choose � such that � < 1=2�.1�ˇ/q=2. In other words, if ˛C1�1=q >1, then let ˇ D 1 and so � < 1=2. If ˛C 1� 1=q � 1, then let ˇ D ˛C 1� 1=q � �and so � < ˛=2 < ˛q=2 � �q=2 for a small enough � > 0. Similarly,

Z

A4\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cp

1X

iDr0

iX

jDi�12.��1=2C.1�ˇ/q=2/.iCj/M.1/

p f .x; y/

C Cp

1X

iDr0

iX

jDi�12.�1=2C.1�ˇ/q=2/.iCj/f .x; y/

� Cp2.��1=2C.1�ˇ/q=2/r0M.1/

p f .x; y/C Cp2.�1=2C.1�ˇ/q=2/r0 f .x; y/ ! 0:

For the set A5, we obtain

Z

A5\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

�r0X

iD1

iX

jDi�1

Z 2iC1=T

2i=T

Z 2jC1=T


!1=p

Z 2iC1=T

2i=T

Z s

s�1=Ts�2q dt ds

!1=q

:

Since

Z 2iC1=T

2i=T

Z s

s�1=Ts�2q dt ds � T�1

�

2i

T

��2qC1;

we concludeZ

A5\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

�r0X

iD1

iX

jDi�1

Z 2iC1=T

2i=T

Z 2jC1=T


!1=p

T�1=q

�

2i

T

��2C1=q

� Cp

r0X

iD1

iX

jDi�12.��1=.2q//.iCj/

2��.iCj/

T2

2iCj

Z 2iC1=T

2i=T

Z 2jC1=T


!1=p

� Cp

r0X

iD1

iX

jDi�12.��1=.2q//.iCj/U.1/

r;p f .x; y/ < C�:

Finally,Z

A5\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cp

1X

iDr0

iX

jDi�12.��1=.2q//.iCj/M.1/

p f .x; y/C Cp

1X

iDr0

iX

jDi�12�.iCj/=.2q/f .x; y/

� Cp2.��1=.2q//r0M.1/

p f .x; y/C Cp2�r0=.2q/f .x; y/ ! 0;

as T ! 1. This finishes the proof of the theorem. �

Proof of Theorem 5.4.19 for d D 2 Taking into account Theorem 5.4.14, theinequality

Z

Sr=2

j f0.x � s/ � f0.x/j j f0.y � t/ � f0.y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt < �

can be proved similarly as in Theorem 5.4.15. Hence we have to estimate the integralZ

S5iD1.Ai\Sc

r=2/

j f0.x � s/ � f0.x/j j f0.y � t/ � f0.y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt:

For small ı > 0 and 1=T < ı let us introduce the sets

B1 WD f.s; t/ W s > r=2; 0 < t � 1=Tg;B2 WD f.s; t/ W s > r=2; 1=T < t � ıg;B3 WD f.s; t/ W s > r=2; ı < t � s � ıg;B4 WD f.s; t/ W s > r=2; s � ı < t � sg:

Then we have to integrate over these four sets. For brevity, we introduce thefunctions

f1.s/ WD f0.x � s/ � f0.x/ and f2.t/ WD f0.y � t/ � f0.y/;

where x and y are fixed. Obviously, f1; f2 2 W.L1; `1/.R/. On B1 we use the factthat f2 is locally bounded at 0 and the estimation (5.4.5) to obtain

Z

B1

j f1.s/j j f2.t/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� CT1�˛1X

iD0.i _ 1/�1�˛

Z iC1

i

Z 1=T

0

j f1.s/j j f2.t/j ds dt

� CT�˛ k f kW.L1;`1/ ! 0;

as T ! 1. Similarly, by (5.4.4),

Z

B2

j f1.s/j j f2.t/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� CT�˛Z 1

r=2

Z ı

1=Ts�1�˛ t�1 j f1.s/j j f2.t/j ds dt

� CT�˛ ln T1X

iD0.i _ 1/�1�˛

Z iC1

ij f1.s/j ds

� CT�˛ ln T k f1kW.L1;`1/ ! 0;

as T ! 1. By (5.4.3),

Z

B4

j f1.s/j j f2.t/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� CN0�1X

iD0.i _ 1/�2

Z iC1

i

Z s

s�ıj f1.s/j j f2.t/j ds dt

C C1X

iDN0

i�2Z iC1

i

Z s

s�ıj f1.s/j j f2.t/j ds dt

� C

f1 f2 1f.s;t/Wr=2<s<N0;s�ı<t�sg

W.L1;`1/C CN�1

0 k f1 f2kW.L1;`1/ :

The second term is less than � if N0 is large enough and the first term is less than �if ı is small enough.

Moreover, by (5.4.7),

Z

B3\A3

j f1.s/j j f2.t/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� CT�˛1X

iD0.i _ 1/�1�˛=2ı�1�˛=2

Z iC1

i

Z 1

ı

j f1.s/j j f2.t/j ds dt

C CT�˛1X

iD1

iX

jD1i�1�˛=2j�1�˛=2

Z iC1

i

Z jC1

jj f1.s/j j f2.t/j ds dt

� CT�˛ k f kW.L1;`1/ ! 0;

as T ! 1. By (5.4.8),

Z

B3\A4

j f1.s/j j f2.t/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� CT�˛1X

iD0

iX

jD0.i _ 1/�1�ˇ. j _ 1/ˇ�˛�1

Z iC1

i

Z s�j

s�j�1j f1.s/j j f2.t/j ds dt

� CT�˛ k f kW.L1;`1/ ! 0;

because we choose ˇ D ˛=2 if 0 < ˛ � 2 and ˇ D 1 if ˛ > 2. �

5.4.2.2 Proof of the Results for q D 1 and d D 2

Recall that we have proved in (5.1.4) and Lemma 5.2.4 that

ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT2; (5.4.10)ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � C.s � t/�1�ˇ.s C t/�1Cˇ; (5.4.11)ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT�˛.s � t/�1�ˇ t�˛Cˇ�1; (5.4.12)ˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ � CT1�˛ t�1�˛; (5.4.13)

where 0 � ˇ � 1.

Proof of Theorem 5.4.15 for q D 1 and d D 2 We use the same notations andFig. 5.20 as for q D 1. The integral (5.4.6) over A1 can be estimated in thesame way. Let � < ˛=2 ^ 1, 2=T < r=2 and Urf .x; y/ < �. By (5.4.11) withˇ D 1=2,

Z

A2\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1

�

2i

T

��3=2 �2i

T

��1=2 Z 2iC1=T

2i=T

Z 1=T

0

j f .x � s; y � t/ � f .x; y/j ds dt

� Cr0X

iD12.��1/i2�� i

�

T2

2i

�Z 2iC1=T

2i=T

Z 1=T

0

j f .x � s; y � t/ � f .x; y/j ds dt

� Cr0X

iD12.��1/iU.1/

r f .x; y/ < C�:

Recall that r0 denotes the largest number i, for which r=2 � 2iC1=T < r. Sinces � t � s=2 on the set A3, we get from (5.4.12) that

Z

A3\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1

i�1X

jD0T�˛

�

2i

T

��1�ˇ �2j

T

��˛Cˇ�1

Z 2iC1=T

2i=T

Z 2jC1=T


� Cr0X

iD1

i�1X

jD02.��ˇ/i2.��˛Cˇ/j

2��.iCj/

�

T2

2iCj

�Z 2iC1=T

2i=T

Z 2jC1=T


� Cr0X

iD1

i�1X

jD02.��ˇ/i2.��˛Cˇ/jU.1/

r f .x; y/ < C�;

where we choose ˇ D ˛=2 if 0 < ˛ � 2 and ˇ D 1 if ˛ > 2.We have t > s=2 on A4, hence (5.4.12) implies

Z

A4\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1

i�1X

jD0T�˛

�

2i

T

��˛Cˇ�1 �2j

T

��1�ˇ

Z 2iC1=T

2i=T

Z s�2j=T


� Cr0X

iD1

i�1X

jD02.��˛Cˇ/i2.��ˇ/j

2��.iCj/

�

T2

2iCj

�Z 2iC1=T

2i=T

Z s�2j=T


� Cr0X

iD1

i�1X

jD02.��˛Cˇ/i2.��ˇ/jU.2/

r f .x; y/ < C�;

where ˇ is chosen as before.

On the set A5, we get from (5.4.13) that

Z

A5\Sr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� Cr0X

iD1T1�˛

�

2i

T

��1�˛ Z 2iC1=T

2i=T

Z s


� Cr0X

iD12.��˛/i2�� i

�

T2

2i

�Z 2iC1=T

2i=T

Z s


� Cr0X

iD12.��˛/iU.2/

r f .x; y/ < C�:

Similarly, we can show that

Z

A2\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

2.��1/iM.1/f .x; y/C C1X

iDr0

2�if .x; y/

� C2.��1/r0M.1/f .x; y/C C2�r0 f .x; y/

� C.Tr/��1M.1/f .x; y/C C.Tr/�1f .x; y/ ! 0

andZ

A3\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

i�1X

jD02.��ˇ/i2.��˛Cˇ/jM.1/f .x; y/C C

1X

iDr0

i�1X

jD02�ˇi2.�˛Cˇ/jf .x; y/

� C2.��ˇ/r0M.1/f .x; y/C C2�ˇr0 f .x; y/ ! 0

andZ

A4\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

i�1X

jD02.��˛Cˇ/i2.��ˇ/jM.2/f .x; y/C C

1X

iDr0

i�1X

jD02.�˛Cˇ/i2�ˇjf .x; y/

� C2.��˛Cˇ/r0M.2/f .x; y/C C2�ˇr0 f .x; y/ ! 0

as T ! 1. Finally,

Z

A5\Scr=2

j f .x � s; y � t/ � f .x; y/jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iDr0

2.��˛/iM.2/f .x; y/C C1X

iDr0

2�˛if .x; y/

� C2.��˛/r0M.2/f .x; y/C C2�˛r0 f .x; y/ ! 0;

as T ! 1. �

5.4.2.3 Proof of the Results for q D 1 and d � 3

We may suppose again that x1 > x2 > : : : > xd > 0. To avoid some technicaldifficulties, we will also suppose in this subsection that x1 �Pd

jD2 xj > 0. If we donot suppose this condition, then the results can be proved with the same ideas.

Lemma 5.4.20 Suppose that (5.1.7) and (5.1.8) are satisfied for some 0 < ˛ < 1.Then

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CTd�jx�11 � � � x�1

j (5.4.14)

for all j D 0; : : : ; d. If in addition x1 � PdjD2 xj > 1=T and xjC1 < 1=T, where

xdC1 D 0, then

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CT�˛Cd�jx�11 � � � x�1

j

0

@x1 �dX

jD2xj

1

A

�˛

(5.4.15)

for all j D 1; : : : ; d.

Proof Recall the inequality (5.2.15):

K1;�T .x/

D �2�dC1��dX

�2;:::;�d�1D˙1

dY

iD1x�1

i

Z 1

0

� 0.t/

0

@soc

0

@tT

0

@

d�1X

jD1�jxj C xd

1

A

1

A � soc

0

@tT

0

@

d�1X

jD1�jxj � xd

1

A

1

A

1

A dt:

Now (5.1.8) with i D 0 implies (5.4.15) for j D d. Estimating soc by 1, weget (5.4.14) for j D d. By Lagrange’s mean value theorem, we obtain that there


exists �d 2 .�xd; xd/ such that

K1;�T .x/ D �2�dC1��d

X

�2;:::;�d�1D˙1

d�1Y

iD1x�1

i

Z 1

0

� 0.t/tTsoc 00

@tT

0

@

d�1X

jD1�jxj � �d

1

A

1

A dt:

This implies (5.4.14) and (5.4.15) for j D d �1. Continuing this procedure we finishthe proof of the lemma. �

If we do not suppose the condition x1 �PdjD2 xj > 0, then let

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

x1 CdX

jD2 jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

have the minimal absolute value betweenˇ

ˇ

ˇx1 CPdjD2 �jxj

ˇ

ˇ

ˇ, i.e.

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

x1 CdX

jD2 jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

x1 CdX

jD2�jxj

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

for all �j D ˙1, where j D ˙1. Then Lemma 5.4.20 can be proved withˇ

ˇ

ˇx1 CPdjD2 jxj

ˇ

ˇ

ˇ instead of x1 � PdjD2 xj on the right-hand side. Indeed, (5.4.14)

and (5.4.15) for j D d can be proved in the same way. To prove the rest of (5.4.15)

suppose thatˇ

ˇ

ˇx1 CPdjD2 jxj

ˇ

ˇ

ˇ > 1=T. Lagrange’s mean value theorem can be

applied, if

x1 Cd�1X

jD2�jxj C xd and x1 C

d�1X

jD2�jxj � xd (5.4.16)

have pairwise the same signs for all �j D ˙1. In this case (5.4.15) holds for j D d�1.If the numbers in (5.4.16) have different signs for some �j D ˙1, then xd > 1=T.Continuing this procedure, we can show (5.4.15).

Proof of Theorem 5.4.15 for q D 1 and d � 3 Similar to the proof for d D 2, wehave to estimate the integral

ˇ

ˇ

ˇ�q;�T f .x/ � f .x/

ˇ

ˇ

ˇ �Z

Rdj f .x � s/ � f .x/j

ˇ

ˇ

ˇKq;�T .s/

ˇ

ˇ

ˇ ds; (5.4.17)

where q D 1;1. For simplicity, we will prove the theorem for d D 3. It can beproved for higher dimensions similarly. As we mentioned earlier, we may supposethat s1 > s2 > s3 > 0 and s1 � s2 � s3 > 0. Then we have to integrate over the sets

A1 WD fs W 0 < s1 � 8=T; 0 < s3 < s2 < s1g ;A2 WD fs W s1 > 8=T; s3 < s2 < 1=T < s1g ;A3 WD fs W s1 > 8=T; s3 < 1=T < s2 < s1g ;A4 WD fs W s1 > 8=T; 1=T < s3 < s2 < s1g :

Let � < ˛=3 ^ 1=2. Since x is a modified strong Lebesgue point of f , we can fixa number r < 1 such that Urf .x/ < �. Let us denote the cube Œ0; r=2�3 by Sr=2 andlet 8=T < r=2. We will integrate the right-hand side of (5.4.17) over the sets

4[

iD1.Ai \ Sr=2/ and

4[

iD1.Ai \ Sc

r=2/:

Since A1 � Sr=2, we have by (5.4.14),

Z

A1

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT3Z 8=T

0

Z 8=T

0

Z 8=T

0

j f .x � s/� f .x/j ds

� CU.1/r f .x/ < C�:

Let us denote by r0 again the largest number i, for which r=2 � 2iC1=T < r. Letus introduce the following sets:

B1 WD fs W 0 < s1 � s2 � s3 < 3=Tg ;B2 WD fs W 3=T < s1 � s2 � s3 < .s1 � s2/=2g ;B3 WD fs W .s1 � s2/=2 _ 3=T < s1 � s2 � s3 < s1 � s2 � 1=Tg ;B4 WD fs W .s1 � s2 � 1=T/ _ 3=T < s1 � s2 � s3 < s1 � s2g

and

C1 WD fs W 0 < s1 � s2 < 1=Tg ;C2 WD fs W 1=T < s1 � s2 < s1=2g ;C3 WD fs W s1=2 < s1 � s2 < s1 � 1=Tg ;C4 WD fs W s1 � 1=T < s1 � s2 < s1g :

On B1 we will use the estimation (5.4.14) while on B2;B3;B4 the estimation (5.4.15).Obviously, A2 \ B1 D ;. On the set A2 \ .B2 [ B3 [ B4/ we have s2; s3 < 1=T andso s1 � s2 � s3 > s1=2. Hence

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT2�˛s�11 .s1 � s2 � s3/

�˛ � CT2�˛s�1�˛1 (5.4.18)

andZ

A2\.B2[B3[B4/\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3T2�˛

�

2i

T

��1�˛ Z 2iC1=T

2i=T

Z 1=T

0

Z 1=T

0


� Cr0X

iD32.��˛/i2�� i

�

T3

2i

�Z 2iC1=T

2i=T

Z 1=T

0

Z 1=T

0

j f .x � s/ � f .x/j ds

� Cr0X

iD32.��˛/iU.1/

r f .x/ < C�:

On A3 \ B1 we have s1 � s2 � s3 < 3=T and s3 < 1=T. Thus s1 � s2 < 4=T ands2 > s1=2. Then

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CTs�11 s�1

2 � CTs�21 (5.4.19)

andZ

A3\B1\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3T

�

2i

T

��2 Z 2iC1=T

2i=T

Z s1

s1�4=T

Z 1=T

0

j f .x � s/ � f .x/j ds

� Cr0X

iD32.��1/i2�� i

�

T3

2i

�Z 2iC1=T

2i=T

Z s1

s1�4=T

Z 1=T

0


� Cr0X

iD32.��1/iUrf .x/ < C�:

On Bi, s1 � s2 > 3=T and so Bi \ C1 D ;, i D 2; 3; 4. If s1 � s2 � s3 > 3=T,s3 < 1=T, then s1 � s2 > 3=T and

s1 � s2 � s3 > s1 � s2 � 1=T > .s1 � s2/=2:

Thenˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT1�˛s�11 s�1

2 .s1 � s2 � s3/�˛ � CT1�˛s�1

1 s�12 .s1 � s2/

�˛ :(5.4.20)

On C2 we have

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT1�˛s�21 .s1 � s2/

�˛ (5.4.21)

andZ

A3\.B2[B3[B4/\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0T1�˛

�

2i

T

��2 �2j

T

��˛

Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z 1=T

0


� Cr0X

iD3

iX

jD02.��1/i2.�C1�˛/j2��.iCj/

�

T3

2iCj

�Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z 1=T

0

j f .x � s/ � f .x/j ds < C�

if 2 � ˛ < 1 and � < 1. If 0 < ˛ < 2, then

2.��1/i2.�C1�˛/j D 2.��˛=2/i2.˛=2�1/i2.�C1�˛/j � 2.��˛=2/i2.��˛=2/j

and soZ

A3\.B2[B3[B4/\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD02.��˛=2/.iCj/Urf .x/ < C�

because � < ˛=2. Inequality (5.4.20) implies

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT1�˛s�1�˛1 s�1

2 (5.4.22)

on C3. Consequently,

Z

A3\.B2[B3[B4/\C3\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0T1�˛

�

2i

T

��1�˛ �2j

T

��1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0

j f .x � s/ � f .x/j ds

� Cr0X

iD3

iX

jD02.��˛=2/.iCj/2��.iCj/

�

T3

2iCj

�

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0

j f .x � s/ � f .x/j ds

� Cr0X

iD3

iX

jD02.��˛=2/.iCj/U.1/

r f .x/ < C�:

On C4 we have s2 < 1=T, which contradicts A3. Consider the set A4 \ B1 and theestimation

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � Cs�11 s�1

2 s�13 : (5.4.23)

Then C1 contradicts s1 � s2 � s3 > 0 and s3 > 1=T and C4 to s2 > 1=T. In otherwords, A4 \ C1 D A4 \ C4 D ;. If s1 � s2 > 6=T, then

s3 > s1 � s2 � 3=T > .s1 � s2/=2

and if 1=T < s1 � s2 < 6=T, then s3 > 1=T > .s1 � s2/=6. Hence on C2,

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � Cs�21 .s1 � s2/

�1 (5.4.24)

andZ

A4\B1\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0

�

2i

T

��2 �2j

T

��1

Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z s1�s2

s1�s2�1=Tj f .x � s/ � f .x/j ds

� Cr0X

iD3

iX

jD02.��1=2/.iCj/2��.iCj/

�

T3

2iCj

�Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z s1�s2

s1�s2�1=Tj f .x � s/ � f .x/j ds < C�:

On C3 we have

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � Cs�21 s�1

2 (5.4.25)

andZ

A4\B1\C3\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0

�

2i

T

��2 �2j

T

��1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2

s1�s2�1=Tj f .x � s/ � f .x/j ds

� Cr0X

iD3

iX

jD02.��1=2/.iCj/2��.iCj/

�

T3

2iCj

�Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2

s1�s2�1=Tj f .x � s/ � f .x/j ds < C�:

On A4 \ .B2 [ B3 [ B4/, we use the estimation

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�11 s�1

2 s�13 .s1 � s2 � s3/

�˛ : (5.4.26)

On B2, s3 > .s1 � s2/=2 and

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�11 s�1

2 .s1 � s2/�1 .s1 � s2 � s3/

�˛ (5.4.27)

� CT�˛s�21 .s1 � s2/

�1 .s1 � s2 � s3/�˛

on B2 \ C2. From this it follows thatZ

A4\B2\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0

jX

kD0T�˛

�

2i

T

��2 �2j

T

��1 �2k

T

��˛

Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z s1�s2�2k=T

s1�s2�2kC1=Tj f .x � s/ � f .x/j ds

� Cr0X

iD3

iX

jD0

jX

kD02.��1/i2� j2.�C1�˛/k2��.iCjCk/

�

T3

2iCjCk

�Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z s1�s2�2k=T

s1�s2�2kC1=Tj f .x � s/ � f .x/j ds < C�

if 2 � ˛ < 1 and � < 1=2. If 0 < ˛ < 2, then

2.��1/i2� j2.�C1�˛/k D 2.��˛=3/i2.˛=3�1/i2� j2.�C1�˛/k

� 2.��˛=3/i2.��˛=3/j2.��˛=3/k

and soZ

A4\B2\C2\Sr=2

j f .x � s/ � f .x/j ˇˇK�T .s/

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0

jX

kD02.��˛=3/.iCjCk/Urf .x/ < C�:

On B2 \ C3,

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�21 s�1

2 .s1 � s2 � s3/�˛

andZ

A4\B2\C3\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0

jX

kD0T�˛

�

2i

T

��2 �2j

T

��1 �2k

T

��˛

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2�2k=T

s1�s2�2kC1=Tj f .x � s/ � f .x/j ds < C�

as before. We obtain from (5.4.26) that

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�11 s�1

2 s�13 .s1 � s2/

�˛ � CT�˛s�21 s�1

3 .s1 � s2/�˛

on B3 \ C2. ThenZ

A4\B3\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0

jX

kD0T�˛

�

2i

T

��2 �2j

T

��˛ �2k

T

��1

Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z 2kC1=T

2k=Tj f .x � s/ � f .x/j ds

� Cr0X

iD3

iX

jD0

jX

kD02.��1/i2.�C1�˛/j2�k2��.iCjCk/

�

T3

2iCjCk

�Z 2iC1=T

2i=T

Z s1�2j=T

s1�2jC1=T

Z 2kC1=T

2k=Tj f .x � s/� f .x/j ds < C�

if 2 � ˛ < 1. The case 0 < ˛ < 2 can be proved as above.On the set B3 \ C3,

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�1�˛1 s�1

2 s�13

andZ

A4\B3\C3\Sr=2

j f .x � s/� f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jD0

jX

kD0T�˛

�

2i

T

��1�˛ �2j

T

��1 �2k

T

��1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


� Cr0X

iD3

iX

jD0

jX

kD02.��˛/i2� j2�k2��.iCjCk/

�

T3

2iCjCk

�Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


� Cr0X

iD3

iX

jD0

jX

kD02.��˛=3/.iCjCk/U.1/

r f .x/ < C�:

Finally, B4 contradicts s3 > 1=T.The integral over Sc

r=2 can be estimated similarly. Basically we write M.1/f .x/Cj f .x/j (resp. Mf .x/C j f .x/j) instead of U.1/

r f .x/ (resp. Urf .x/) in each estimation.For example,

Z

A2\.B2[B3[B4/\Scr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iDr0

2.��˛/i2�� i

�

T3

2i

�Z 2iC1=T

2i=T

Z 1=T

0

Z 1=T

0

j f .x � s/ � f .x/j ds

� C1X

iDr0

2.��˛/iM.1/f .x/C C1X

iDr0

2�˛ij f .x/jj

� C2.��˛/r0M.1/f .x/C C2�˛r0 j f .x/j� C.Tr/��˛M.1/f .x/C C.Tr/�˛ j f .x/j ! 0

as T ! 1. This completes the proof of the theorem. �Proof of Theorem 5.4.17 for d � 3 We have to integrate the integral in (5.4.17) onthe same sets as before. Now let � < ˛=3 ^ 1=4 ^ 1=.12q/, where 1=p C 1=q D1. Since x is a modified p-Lebesgue point of f , we can fix a number r such thatU.1/

r;p f .x/ < �. Taking into account (5.4.9), we can estimate the integral (5.4.17) onthe sets A1, A2, A3 \ .B2 [ B3 [ B4/ \ C3 and A4 \ B3 \ C3 in the same way asbefore.

By (5.4.19) and Hölder’s inequality,

Z

A3\B1\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

�r0X

iD3

iX

jDi�1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0

j f .x � s/ � f .x/j ˇˇK�T .s/

ˇ

ˇ 1A3\B1 .s/ ds

� Cr0X

iD3

iX

jDi�1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0

j f .x � s/ � f .x/jp ds

!1=p

Z 2iC1=T

2i=T

Z s1

s1�4=T

Z 1=T

0

Tqs�2q1 1A3\B1 .s/ ds

!1=q

and soZ

A3\B1\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jDi�1T1�2=q

�

2i

T

��2C1=q

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0


!1=p

� Cp

r0X

iD3

iX

jDi�12.��1=2q/.iCj/

2��.iCj/

T3

2iCj

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0


!1=p

� Cp

r0X

iD3

iX

jDi�12.��1=2q/.iCj/U.1/

r;p f .x/ < C�:

Similarly,

Z

A3\.B2[B3[B4/\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jDi�1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0


!1=p

Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z 1=T

0

T.1�˛/qs�2q1 .s1 � s2/

�˛q 1A3\.B2[B3[B4/\C2 .s/ ds

!1=q

:

If q > 1=.2˛/, then

Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z 1=T

0

T.1�˛/qs�2q1 .s1 � s2/

�˛q ds

�Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z 1=T

0

T.1�˛/qs�2qC1=21 .s1 � s2/

�1=2�˛q ds

� CT.1�˛/q�1�

2i

T

��2qC3=2 �1

T

�1=2�˛q

� C

�

T3q�3

2i.2q�2/

�

2�i=2

and soZ

A3\.B2[B3[B4/\C2\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cp

r0X

iD3

iX

jDi�12.��1=6q/.iCj/

2��.iCj/

T3

2iCj

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0


!1=p

< C�:

If q < 1=˛, then

Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z 1=T

0

T.1�˛/qs�2q1 .s1 � s2/

�˛q ds

� CT.1�˛/q�1Z 2iC1=T

2i=Ts�2q1

�

s1 � 2i�1

T

��˛qC1ds

� CT.1�˛/q�1�

2i

T

��˛qC1 �2i

T

��2qC1

� C

�

T3q�3

2i.2q�2/

�

2�i˛q:

andZ

A3\.B2[B3[B4/\C2\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cp

r0X

iD3

iX

jDi�12.��˛=2/.iCj/

2��.iCj/

T3

2iCj

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 1=T

0


!1=p

< C�:

It follows from (5.4.23) that

jK1;�T .s/j � Cs�1

1 s�12 s�ˇ

3 sˇ�13 � Cs�2

1 .s1 � s2/�ˇsˇ�1

3

on the set A4 \ B1 \ C2, where 0 � ˇ. Since s1 � s2 � 3=T < s3 < s1 � s2, we have

Z

A4\B1\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jDi�1

jX

kD1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T

2k=Tj f .x � s/� f .x/jp ds

!1=p

Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z 2kC1=T

2k=Ts�2q1 .s1 � s2/

�ˇq s.ˇ�1/q3 1A4\B1\C2 .s/ ds

!1=q

� Cp

r0X

iD3

iX

jDi�1

jX

kD1T�1=q

�

2i

T

��2C1=q �2k

T

�ˇ�1 �1

T

��ˇC1=q

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

;

whenever ˇ > 1=q. Then

T�1=q

�

2i

T

��2C1=q �2k

T

�ˇ�1 �1

T

��ˇC1=q

� T3�3=q2i.�2C2=q/2k.�1C1=q/2�i=q2k.ˇ�1=q/

� T3�3=q2.iCjCk/.�1C1=q/2i.ˇ�2=q/;

where we set for example ˇ D 3=.2q/. We conclude

Z

A4\B1\C2\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cp

r0X

iD3

iX

jDi�1

jX

kD12.��1=6q/.iCjCk/

2��.iCjCk/

T3

2iCjCk

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

< C�:

On A4 \ B1 \ C3 we haveZ

A4\B1\C3\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

i�1X

jD1

jX

kD1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T

2k=Tj f .x � s/ � f .x/jp ds

!1=p

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2

s1�s2�1=Ts�2q1 s�q

2 1A4\B1\C3 .s/ ds

!1=q

� Cp

r0X

iD3

i�1X

jD1

jX

kD1T�1=q

�

2i

T

��2C1=q �2j

T

��1C1=q

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

� Cp

r0X

iD3

i�1X

jD1

jX


2��.iCjCk/

T3

2iCjCk

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

< C�:

On A4 \ B2 \ C2 we obtain

jK1;�T .s/j � CT�˛s�1

1 s�12 s�1

3 .s1 � s2 � s3/�˛

� CT�˛s�21 .s1 � s2/

�3=2qs3=2q�13 .s1 � s2 � s3/

�˛ :

ThenZ

A4\B2\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jDi�1

jX

kD1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z 2kC1=T

2k=T

T�˛qs�2q1 .s1 � s2/

�3=2 s3=2�q3 .s1 � s2 � s3/

�˛q 1A4\B2\C2 .s/ ds

!1=q

:

If q > 3=.4˛/, then

�

2k

T

�3=2�q Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z s1�s2�3=T

.s1�s2/=2T�˛qs�2q

1 .s1 � s2/�3=2 .s1 � s2 � s3/

�˛q ds

��

2k

T

�3=2�q Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z s1�s2�3=T

.s1�s2/=2

T�˛qs�2qC1=41 .s1 � s2/

�3=2 .s1 � s2 � s3/�1=4�˛q ds

� C

�

2k

T

�3=2�q

T�˛q

�

2i

T

��2qC5=4 �1

T

��1=2 �1

T

�3=4�˛q

� CT3q�32i.�2qC2/2k.�qC1/2�i=4

and soZ

A4\B2\C2\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cp

r0X

iD3

iX

jDi�1

jX


2��.iCjCk/

T3

2iCjCk

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

< C�:

If 1=.4˛/ < q < 1=˛, then

�

2k

T

�3=2�q Z 2iC1=T

2i=T

Z s1�1=T

2i�1=T

Z s1�s2�3=T

.s1�s2/=2

T�˛qs�2q1 .s1 � s2/

�3=2 .s1 � s2 � s3/�˛q ds

��

2k

T

�3=2�q Z 2iC1=T

2i=T

Z s1�1=T

2i�1=TT�˛qs�2qC1=4

1 .s1 � s2/�1=2�1=4�˛q ds1 ds2

� C

�

2k

T

�3=2�q

T�˛q

�

2i

T

��2qC5=4 �1

T

�1=4�˛q

� CT3q�32i.�2qC2/2k.�qC1/2�i=4

and the estimation can be finished as above.

If q < 1=.2˛/, then

�

2k

T

�3=2�q Z 2iC1=T

2i=T

Z s1�1=T

2i�1=TT�˛qs�2q

1 .s1 � s2/�1=2�˛q ds1 ds2

��

2k

T

�3=2�q Z 2iC1=T

2i=TT�˛qs�2q

1

�

s1 � 2i�1

T

�1=2�˛q

ds1

� C

�

2k

T

�3=2�q

T�˛q

�

2i

T

��2qC1 �2i

T

�1=2�˛q

� CT3q�32i.�2qC2/2k.�qC1/2�i˛q

andZ

A4\B2\C2\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cp

r0X

iD3

iX

jDi�1

jX

kD12.��˛=3/.iCjCk/

2��.iCjCk/

T3

2iCjCk

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

< C�:

Integrating on A4 \ B2 \ C3, we obtain

Z

A4\B2\C3\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

i�1X

jD1

jX

kD1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2�3=T

.s1�s2/=2

T�˛qs�2q1 s�q

2 .s1 � s2 � s3/�˛q 1A4\B2\C3 .s/ ds

!1=q

:

If q > 3=.4˛/, then

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2�3=T

.s1�s2/=2T�˛qs�2q

1 s�q2 .s1 � s2 � s3/

�˛q ds

�Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2�3=T

.s1�s2/=2

T�˛qs�2qC1=41 s�q

2 .s1 � s2 � s3/�1=4�˛q ds

� CT�˛q

�

2i

T

��2qC5=4 �2j

T

��qC1 �1

T

�3=4�˛q

� CT3q�32i.�2qC2/2j.�qC1/2�3i=4:

If q < 1=˛, then

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z s1�s2�3=T

.s1�s2/=2T�˛qs�2q

1 s�q2 .s1 � s2 � s3/

�˛q ds

�Z 2iC1=T

2i=T

Z 2jC1=T

2j=TT�˛qs�2q�˛qC1

1 s�q2 ds1 ds2

� CT�˛q

�

2i

T

��2q�˛qC2 �2j

T

��qC1

� CT3q�32i.�2qC2/2j.�qC1/2�i˛q:

In both casesZ

A4\B2\C3\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds < C�:

Finally,

Z

A4\B3\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� Cr0X

iD3

iX

jDi�1

jX

kD1

Z 2iC1=T

2i=T

Z 2jC1=T

2j=T

Z 2kC1=T


!1=p

Z 2iC1=T

2i=T

Z s1�1=T

s1=2

Z 2kC1=T

2k=TT�˛qs�2q

1 s�q3 .s1 � s2/

�˛q 1A4\B3\C2 .s/ ds

!1=q

:

If q > 3=.4˛/, then

Z 2iC1=T

2i=T

Z s1�1=T

s1=2

Z 2kC1=T

2k=TT�˛qs�2q

1 s�q3 .s1 � s2/

�˛q ds

�Z 2iC1=T

2i=T

Z s1�1=T

s1=2

Z 2kC1=T

2k=TT�˛qs�2qC1=4

1 s�q3 .s1 � s2/

�1=4�˛q ds

� CT�˛q

�

2i

T

��2qC5=4 �2k

T

��qC1 �1

T

�3=4�˛q

� CT3q�32i.�2qC2/2k.�qC1/2�3i=4:

If q < 1=˛, then

Z 2iC1=T

2i=T

Z s1�1=T

s1=2

Z 2kC1=T

2k=TT�˛qs�2q

1 s�q3 .s1 � s2/

�˛q ds

�Z 2iC1=T

2i=T

Z 2kC1=T

2k=TT�˛qs�2q�˛qC1

1 s�q3 ds1 ds3

� CT�˛q

�

2i

T

��2q�˛qC2 �2k

T

��qC1

� CT3q�32i.�2qC2/2k.�qC1/2�i˛q

and in both casesZ

A4\B3\C2\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds < C�:

The integral over Scr=2 can be estimated similarly. �

Proof of Theorem 5.4.19 for d � 3 For simplicity, we will prove the theorem ford D 3, only. We can prove in the same way as above that

Z

Sr=2

3Y

jD1

ˇ

ˇ f0.xj � sj/ � f0.xj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds < �

if r is small enough and T is large enough. We will show that

limT!1

Z

Scr=2

3Y

jD1

ˇ

ˇ f0.xj � sj/� f0.xj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds D 0: (5.4.28)

To this end let us introduce the sets

D1 WD fs W s1 > r=2; 0 < s3 < s2 < 1=Tg ;D2 WD fs W s1 > r=2; 0 < s3 < 1=T < s2 < ıg ;D3 WD fs W s1 > r=2; 1=T < s3 < s2 < ıg ;D4 WD fs W s1 > r=2; 0 < s3 < 1=T < ı < s2g ;D5 WD fs W s1 > r=2; 1=T < s3 < ı < s2g ;D6 WD fs W s1 > r=2; 1=T < ı < s3 < s2g

and

E1 WD fs W 0 < s1 � s2 � s3 < 1=Tg ;E2 WD fs W 1=T < s1 � s2 � s3 < ıg ;E3 WD fs W ı < s1 � s2 � s3 < .s1 � s2/=2g ;E4 WD fs W .s1 � s2/=2 _ ı < s1 � s2 � s3 < s1 � s2 � ıg ;E5 WD fs W .s1 � s2 � ı/ _ ı < s1 � s2 � s3 < s1 � s2g

and

F1 WD fs W 0 < s1 � s2 < ıg ;F2 WD fs W ı < s1 � s2 < s1=2g ;F3 WD fs W s1=2 < s1 � s2 < s1 � ıg ;F4 WD fs W s1 � ı < s1 � s2 < s1g

for a given small ı > 0 and large T. Then we have to estimate the integral in (5.4.28)for Di, i D 1; : : : ; 6. We introduce again the functions

fj.sj/ WD f0.xj � sj/ � f0.xj/ . j D 1; 2; 3/;

where x is fixed. Obviously, f1; f2; f3 2 W.L1; `1/.R/ and they are locally boundedat 0. Observe that Di \ Ej D ; for i D 1; 2; 3 and j D 1; 2. On the set D1 weuse (5.4.18) and the local boundedness of f2 and f3 to get

Z

D1\.E3[E4[E5/

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT2�˛Z 1

r=2

Z 1=T

0

Z 1=T

0

s�1�˛1

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds


� CT�˛1X

iD0.i _ 1/�1�˛

Z iC1

ij f1.s1/j ds1

� CT�˛ k f kW.L1;`1/ ! 0;

as T ! 1. On D2, we get by (5.4.22) that

Z

D2\.E3[E4[E5/

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT1�˛Z 1

r=2

Z ı

1=T

Z 1=T

0

s�1�˛1 s�1

2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛ ln T1X

iD0.i _ 1/�1�˛

Z iC1

ij f1.s1/j ds1

� CT�˛ ln T k f kW.L1;`1/ ! 0:

Similarly,

Z

D3\.E3[E4[E5/

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z ı

1=T

Z ı

1=Ts�1�˛1 s�1

2 s�13

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛ ln2 T k f kW.L1;`1/ ! 0:

By (5.4.19),

Z

D4\E1

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds (5.4.29)

� CTZ 1

r=2

Z s1

s1�ı

Z 1=T

0

s�21

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CN0�1X

iD0.i _ 1/�2

Z iC1

i

Z s1

s1�ıj f1.s1/j j f2.s2/j ds1 ds2

C C1X

iDN0

i�2Z iC1

i

Z s1

s1�ıj f1.s1/j j f2.s2/j ds1 ds2

� C

. f1 f2/1f.s1;s2/W0<s1<N0;s1�ı<s2�s1g

W.L1;`1/

C CN�10 k f1 f2kW.L1;`1/ :

The second term is less than � if N0 is large enough and the first term is less than �if ı is small enough. The set D4 \ E2 can be handled in the same way.

On D4 \ .E3 [ E4 [ E5/ we have s1 � s2 > ı. Thus .E3 [ E4 [ E5/ \ F1 D ;.By (5.4.20),

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT1�˛s�21 .s1 � s2/

�˛ � CT1�˛s�1��1 .s1 � s2/

�1�˛C�

on F2, where � is chosen such that 0 < � < 1 ^ ˛. Hence

Z

D4\.E3[E4[E5/\F2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT1�˛Z 1

r=2

Z s1�ı

s1=2

Z 1=T

0

s�1��1 .s1 � s2/

�1�˛C�3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛1X

iD0.i _ 1/�1��

X

0� j<.iC1/=2. j _ 1/�1�˛C�

Z iC1

i

Z s1�j

s1�j�1j f1.s1/j j f2.s2/j ds1 ds2

� CT�˛ k f k2W.L1;`1/ ! 0:

On F3, (5.4.20) implies that

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT1�˛s�1�˛1 s�1

2 � CT1�˛s�1�˛C�1 s�1��

2

and

Z

D4\.E3[E4[E5/\F3

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT1�˛Z 1

r=2

Z s1=2

ı

Z 1=T

0

s�1�˛C�1 s�1��

2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛1X

iD0.i _ 1/�1�˛C� X

0� j<.iC1/=2. j _ 1/�1��

Z iC1

i

Z jC1

jj f1.s1/j j f2.s2/j ds1 ds2

� CT�˛ k f k2W.L1;`1/ ! 0:

Observe that s2 > ı contradicts F4.Consider the set D5\E1. If s1�s2 > 2=T, then s3 > s1�s2�1=T > .s1�s2/=2 and

if s1�s2 < 2=T, then s3 > 1=T > .s1�s2/=2. Observe that s1�s2 < 1=T Cı < 2ı.Hence

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � Cs�11 s�1

2 s�13 � Cs�2

1 .s1 � s2/�1=2s�1=2

3 :

Since s3 < ı, we can integrate in s3 to obtain

Z

D5\E1

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CZ 1

r=2

Z s1

s1�2ı

Z s1�s2

s1�s2�1=Ts�21 .s1 � s2/

�1=2s�1=23

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CZ 1

r=2

Z s1

s1�2ıs�21 j f1.s1/j j f2.s2/j ds1 ds2

� C

. f1 f2/1f.s1;s2/W0<s1<N0;s1�2ı<s2�s1g

W.L1;`1/

C CN�10 k f1 f2kW.L1;`1/

as in (5.4.29). Similarly, s1 � s2 < 2ı holds as well on D5 \ E2. If s1 � s2 � s3 > s3,then

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�11 s�1

2 s�13 .s1 � s2 � s3/

�˛ � CT�˛s�21 s�1�˛

3 (5.4.30)

and so

Z

D5\E2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1

s1�2ı

Z ı

1=Ts�21 s�1�˛

3

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CZ 1

r=2

Z s1

s1�2ıs�21 j f1.s1/j j f2.s2/j ds1 ds2 < �

as before. If s3 > s1 � s2 � s3, then

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�21 .s1 � s2 � s3/

�1�˛

and

Z

D5\E2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1

s1�2ı

Z s1�s2�1=T

s1�s2�ıs�21 .s1 � s2 � s3/

�1�˛3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CZ 1

r=2

Z s1

s1�2ıs�21 j f1.s1/j j f2.s2/j ds1 ds2 < �:

If s1 � s2 < 2ı, then the integral on the set D5 \ .E3 [ E4 [ E5/ can be estimatedin the same way. If s1 � s2 > 2ı then D5 \ E3 D ; because s3 > .s1 � s2/=2 ands3 < ı on this set. Writing 1=T instead of ı in the sets Ei .i D 4; 5/, we obtain thedefinition of the sets E0

i . On E04, we have by (5.4.30) that

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ � CT�˛s�11 s�1

2 s�13 .s1 � s2/

�˛

and

Z

D5\E0

4\F2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1�ı

s1=2

Z ı

1=Ts�21 s�1

3 .s1 � s2/�˛

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛ ln T1X

iD0.i _ 1/�1��

X

0� j<.iC1/=2. j _ 1/�1�˛C�

Z iC1

i

Z s1�j

s1�j�1j f1.s1/j j f2.s2/j ds1 ds2

� CT�˛ ln T k f1kW.L1;`1/ k f2kW.L1;`1/ ! 0

and

Z

D5\E0

4\F3

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1=2

ı

Z ı

1=Ts�1�˛1 s�1

2 s�13

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛ ln T1X

iD0.i _ 1/�1�˛C� X

0� j<.iC1/=2. j _ 1/�1��

Z iC1

i

Z jC1

jj f1.s1/j j f2.s2/j ds1 ds2

� CT�˛ ln T k f1kW.L1;`1/ k f2kW.L1;`1/ ! 0:

Moreover, E05 contradicts D5, more exactly, to s3 > 1=T.

On D6, s3 > ı and s1 � s2 � s3 > 0, thus s1 � s2 > ı. By (5.4.24),

Z

D6\E1\F2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CZ 1

r=2

Z s1�ı

s1=2

Z s1�s2

s1�s2�1=Ts�21 .s1 � s2/

�13Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� C1X

iD0.i _ 1/�3=2

X

0� j<.iC1/=2. j _ 1/�3=2

Z iC1

i

Z s1�j

s1�j�1

Z s1�s2

s1�s2�1=Tj f1.s1/j j f2.s2/j j f3.s3/j ds1 ds2 ds3

� C

. f1 f2 f3/1fsW0<s1<N0;s1=2<s2�s1�ı;s1�s2�ı<s3<s1�s2g

W.L1;`1/

C CN�1=20 k f1 f2 f3kW.L1;`1/ ;

which is small enough if N0 is large and ı is small enough. On D6\ E1\ F3, we usethe estimation (5.4.25) to obtain

Z

D6\E1\F3

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CZ 1

r=2

Z s1=2

ı

Z s1�s2

s1�s2�1=Ts�21 s�1

2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� C

. f1 f2 f3/1fsW0<s1<N0;ı<s2�s1=2;s1�s2�ı<s3<s1�s2g

W.L1;`1/

C CN�1=20 k f1 f2 f3kW.L1;`1/ < �;

as just before.

On D6\E2, s1� s2 > ı again. If s1� s2 > 2ı, then s3 > s1� s2�ı > .s1� s2/=2and if s1 � s2 < 2ı, then s3 > ı > .s1 � s2/=2. This case can be handled in the sameway as the set D6 \ E1. On D6 \ E3, we apply (5.4.27),

Z

D6\E3\F2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1�ı

s1=2

Z s1�s2�ı

.s1�s2/=2s�21 .s1 � s2/

�1 .s1 � s2 � s3/�˛

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛1X

iD0.i _ 1/�1�

X

0� j<.iC1/=2. j _ 1/�1�

X

0�k<. jC1/=2.k _ 1/�1�˛C2

Z iC1

i

Z s1�j

s1�j�1

Z s1�s2�k

s1�s2�k�1j f1.s1/j j f2.s2/j j f3.s3/j ds1 ds2 ds3

� CT�˛ k f1kW.L1;`1/ k f2kW.L1;`1/ k f3kW.L1;`1/ ! 0;

where 0 < < 1=2^ ˛=2. Similarly,

Z

D6\E3\F3

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1=2

ı

Z s1�s2�ı

.s1�s2/=2s�21 s�1

2 .s1 � s2 � s3/�˛

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛1X

iD0.i _ 1/�1�

X

0� j<.iC1/=2. j _ 1/�1�

X

0�k<. jC1/=2.k _ 1/�1�˛C2

Z iC1

i

Z jC1

j

Z s1�s2�k

s1�s2�k�1j f1.s1/j j f2.s2/j j f3.s3/j ds1 ds2 ds3

� CT�˛ k f1kW.L1;`1/ k f2kW.L1;`1/ k f3kW.L1;`1/ ! 0:

Moreover, on D6 \ E4 we use the first inequality:

Z

D6\E4\F2

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1�ı

s1=2

Z .s1�s2/=2

ı

s�21 .s1 � s2/

�˛ s�13

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛1X

iD0.i _ 1/�1�

X

0� j<.iC1/=2. j _ 1/�1�˛C2 X

0�k<. jC1/=2.k _ 1/�1�

Z iC1

i

Z s1�j

s1�j�1

Z kC1

kj f1.s1/j j f2.s2/j j f3.s3/j ds1 ds2 ds3

� CT�˛ k f1kW.L1;`1/ k f2kW.L1;`1/ k f3kW.L1;`1/ ! 0

and

Z

D6\E4\F3

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ

ˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛Z 1

r=2

Z s1=2

ı

Z .s1�s2/=2

ı

s�1�˛1 s�1

2 s�13

3Y

jD1

ˇ

ˇ fj.sj/ˇ

ˇ ds

� CT�˛1X

iD0.i _ 1/�1�˛=2

X

0� j<.iC1/=2. j _ 1/�1�˛=2

X

0�k<. jC1/=2.k _ 1/�1�˛=2

Z iC1

i

Z jC1

j

Z kC1

kj f1.s1/j j f2.s2/j j f3.s3/j ds1 ds2 ds3

� CT�˛ k f1kW.L1;`1/ k f2kW.L1;`1/ k f3kW.L1;`1/ ! 0:

Since D6 \ E5 D ;, the proof of the theorem is complete. �

5.4.2.4 Proof of the Results for q D 1 and d � 3

Of course, we suppose again that x1 > x2 > : : : > xd > 0. Recall that the sequence.il; jl/ 2 I was defined in Definition 4.4.3.

Lemma 5.4.21 For all j D 1; : : : ; d � 1,

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CTd�1�jX

.il; jl/2Ix�1

i1

jY

lD1.xil � xjl/

�1: (5.4.31)

Proof In this proof let us write D1;dt instead of the d-dimensional triangular Dirichlet

kernel D1t . Since by Lemma 4.4.4,

ˇ

ˇD1;dt .x/

ˇ

ˇ �X

.il; jl/2I

d�1Y

lD1.x2il � x2jl/

�1 ˇˇGt.x

2id�1/ � Gt.x

2jd�1/ˇ

ˇ

� CX

.il; jl/2I

d�1Y

lD1.x2il � x2jl/

�1 �xd�2id�1

C xd�2jd�1

�

� CX

.il; jl/2Ix�1

i1

d�1Y

lD1.xil � xjl/

�1; (5.4.32)

we obtain by (5.1.9) that

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ �Z 1

0

ˇ

ˇ� 0.t/ˇ

ˇ

ˇ

ˇ

ˇD1;dtT .x/

ˇ

ˇ

ˇ dt � CX

.il; jl/2Ix�1

i1

d�1Y

lD1.xil � xjl /

�1;

which is exactly what we intended to prove for j D d � 1. Next observe that

D1;dt .x/ D 2d

Z

.0;1/d1fjvj�tg cos.x1v1/ � � � cos.xdvd/ dv

D 2dZ t

0

Z t�vd

0

Z t�vd�vd�1

0

: : :

Z t�vd�:::�v2

0

cos.xdvd/ � � � cos.x1v1/ dv

D 2

Z t

0

cos.xdvd/D1;d�1t�vd

.x1; : : : ; xd�1/ dvd:

By (5.4.32),

ˇ

ˇD1;dt .x/

ˇ

ˇ � CtX

.il; jl/2Ix�1

i1

d�2Y

lD1.xil � xjl/

�1

and so

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CZ 1

0

ˇ

ˇ� 0.t/ˇ

ˇ

ˇ

ˇ

ˇD1;dtT .x/

ˇ

ˇ

ˇ dt � CTX

.il; jl/2Ix�1

i1

d�2Y

lD1.xil � xjl/

�1;

which yields (5.4.31) for j D d � 2. Continuing this procedure, we can easily finishthe proof. �

Lemma 5.4.22 Suppose that (5.1.7) and (5.1.8) are satisfied for some 0 < ˛ < 1.If xd > 1=T, then

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CTd�1�j�˛ X

.il; jl/2Ix�1

i1x�˛

j1

jY

lD1.xil � xjl/

�1 (5.4.33)


for all j D 1; : : : ; d � 1 andˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CTd�1�˛x�1�˛j1 :

Proof By Lemma 4.4.4, we have

K1;�T .x/ D �

Z 1

0

� 0.t/DtT .x/ dt (5.4.34)

D CX

.il; jl/2I.�1/id�1

d�1Y

lD1.x2il � x2jl/

�1

Z 1

0

� 0.t/�

GtT.x2id�1/ � GtT.x

2jd�1/�

dt:

Taking into account (5.1.8), we conclude

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CX

.il; jl/2I

d�1Y

lD1.x2il � x2jl/

�1

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/�

xd�2id�1

soc .tTxid�1 / � xd�2jd�1

soc .tTxjd�1 /�

dt

ˇ

ˇ

ˇ

ˇ

� CT�˛ X

.il; jl/2I

d�1Y

lD1.x2il � x2jl/

�1 �xd�2�˛id�1

C xd�2�˛jd�1

�

� CT�˛ X

.il; jl/2Ix�1

i1

d�1Y

lD1.xil � xjl/

�1 �x�˛id�1

C x�˛jd�1

�

� CT�˛ X

.il; jl/2Ix�1

i1x�˛

j1

d�1Y

lD1.xil � xjl/

�1:

Lagrange’s mean value theorem and (5.4.34) imply

K1;�T .x/ D C

X

.il; jl/2I.�1/id�1

d�2Y

lD1.x2il � x2jl/

�1Z 1

0

� 0.t/G0tT.�

2/ dt

and

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � CX

.il; jl/2I

d�2Y

lD1.x2il � x2jl/

�1

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/

�d�4soc .tT�/C �d�3tTsoc 0.tT�/�

dt

ˇ

ˇ

ˇ

ˇ

� CX

.il; jl/2I

d�2Y

lD1.x2il � x2jl/

�1

T�˛�d�4�˛ C T1�˛�d�3�˛�

� CT1�˛X

.il; jl/2Ix�1

i1

d�2Y

lD1.xil � xjl/

�1��˛

� CT1�˛X

.il; jl/2Ix�1

i1x�˛

j1

d�2Y

lD1.xil � xjl/

�1;

where xjd�1 < � < xid�1 . Continuing this process we obtain the inequality for j Dd � 3; : : : ; 1. In the last step we get that

K1;�T .x/ D C

Z 1

0

� 0.t/G.d�1/tT .�2/ dt

and

ˇ

ˇ

ˇK1;�T .x/

ˇ

ˇ

ˇ � Cd�1X

iD0

ˇ

ˇ

ˇ

ˇ

Z 1

0

� 0.t/� i�d.tT/isoc .i/.tT�/ dt

ˇ

ˇ

ˇ

ˇ

� Cd�1X

iD0Ti�˛� i�d�˛

� CTd�1�˛x�1�˛j1 ;


Proof of Theorem 5.4.15 for q D 1 and d � 3 Similar to the proof for q D 1, wewill estimate the integral (5.4.17) on the following sets. We may assume again thats1 > : : : > sd > 0. Let

A0 WD fs W 8=T � s1 > : : : > sd > 0g ;A1 WD ˚

s W s1 > 8=T; sil � sjl > 2=T; 8 l D 1; : : : d � 1; i 2 I� ;Aj WD ˚

s W s1 > 8=T; sil � sjl > 2=T; 8 l D 1; : : : d � j; i 2 I;9 i 2 I; sid�jC1

� sjd�jC1< 2=T

�

;

B WD fs W 0 < sd � 1=Tg ;where j D 2; : : : ; d�1. We will integrate the right-hand side of (5.4.17) over the sets

d�1[

iD0.Ai \B\Sr=2/;

d�1[

iD0.Ai \B\Sc

r=2/;

d�1[

iD0.Ai \Bc \Sr=2/;

d�1[

iD0.Ai \Bc \Sc

r=2/;

where Sr=2 denotes the cube Œ0; r=2�d and let 8=T < r=2.

We fix a number r < 1 such that Urf .x/ < �. The integral on the set A0 can beestimated in the same way as in the proof for q D 1. Let � < ˛=d ^ 1=d. On theset A1 \ B we use (5.4.31) with j D d � 1 to obtain

Z

A1\B\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds (5.4.35)

� CX

.il; jl/2I

Z

A1\B\Sr=2

j f .x � s/ � f .x/j s�1i1

d�1Y

lD1.sil � sjl /

�1 ds

� CX

.il; jl/2I

Z

A1\B\Sr=2

j f .x � s/ � f .x/j s�2i1

d�1Y

lD2.sil � sjl /

�1 ds:

To integrate the right-hand side, let i0d D d D j1 and we will consider the integralR 1=T0

� � � dsi0d. Next, let i01 D i1 D 1 and the integral

r0X

k1D3

Z 2k1C1=T

2k1 =T� � � s�2

i1 dsi01

will be computed. Here r0 denotes again the largest number i, for which r=2 �2iC1=T < r. If i2 D i1 and j2 D j1 � 1, then let i02 D j2 and we consider

k1X

k2D0

Z si2�2k2 =T

si2�2k2C1=T� � � .si2 � sj2 /

�1 dsi02:

If i2 D i1 C 1 and j2 D j1, then let i02 D i2 and we consider

k1X

k2D0

Z sj2C2k2C1=T

sj2C2k2 =T� � � .si2 � sj2 /

�1 dsi02:

For .si3 � sj3 /�1 we have the following two cases. If i3 D i2 and j3 D j2 � 1, then

let i03 D j3 and we consider

k1X

k3D0

Z si3�2k3 =T

si3�2k3C1=T� � � .si3 � sj3 /

�1 dsi03:

If i3 D i2 C 1 and j3 D j2, then let i03 D i3 and we consider

k1X

k3D0

Z sj3C2k3C1=T

sj3C2k3 =T� � � .si3 � sj3 /

�1 dsi03:

Continuing this process we will integrate over a parallelepiped P2k1 =T;:::;2kd =T withside lengths 2k1C1=T; : : : ; 2kdC1=T, where kd D 1. We conclude

Z

A1\B\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CX

.il; jl/2I

r0X

k1D3

k1X

k2D0� � �

k1X

kd�1D0

�

2k1

T

��2 d�1Y

lD2

�

2kl

T

��1

Z

P2k1 =T;:::;2kd =T

j f .x � s/ � f .x/j ds

� CX

.il; jl/2I

r0X

k1D3

k1X

k2D0� � �

k1X

kd�1D02.��1=d/kkk1

2��kkk1 1ˇ

ˇP2k1 =T;:::;2kd =T

ˇ

ˇ

Z

P2k1 =T;:::;2kd =T

j f .x � s/ � f .x/j ds

and soZ

A1\B\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CX

.il; jl/2I

r0X

k1D3

k1X

k2D0� � �

k1X

kd�1D02.��1=d/kkk1Urf .x/ < C�:

Similarly,

Z

A1\B\Scr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CX

.il; jl/2I

1X

k1Dr0

k1X

k2D0� � �

k1X

kd�1D0

�

2k1

T

��2 d�1Y

lD2

�

2kl

T

��1

Z

P2k1 =T;:::;2kd =T


� CX

.il; jl/2I

1X

k1Dr0

k1X

k2D0� � �

k1X

kd�1D02.��1=d/kkk1

2��kkk1 1ˇ

ˇP2k1 =T;:::;2kd =T

ˇ

ˇ

Z

P2k1 =T;:::;2kd =T

j f .x � s/ � f .x/j ds;

thusZ

A1\B\Scr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CX

.il; jl/2I

1X

k1Dr0

k1X

k2D0� � �

k1X

kd�1D02.��1=d/kkk1Mf .x/

C CX

.il; jl/2I

1X

k1Dr0

k1X

k2D0� � �

k1X

kd�1D02�kkk1=dj f .x/j

� C1X

k1Dr0

2.��1=d/k1Mf .x/C C1X

k1Dr0

2�k1=dj f .x/j

� C2.��1=d/r0Mf .x/C C2�r0=dj f .x/j ! 0

as T ! 1. Note that here kd D 1.On the set A2\B we use (5.4.31) with j D d�2. Suppose that there exists exactly

one difference sid�1 � sjd�1 for which sid�1 � sjd�1 < 2=T, say sd�1� sd < 2=T. Then

Z

A2\B\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds (5.4.36)

� CTX

.il; jl/2I

Z

A2\B\Sr=2

j f .x � s/ � f .x/j s�2i1

d�2Y

lD2.sil � sjl/

�1 ds:

Let i0d D d D j1, i0d�1 D d � 1 D id�1 and consider the integrals

Z 1=T

0

� � � dsi0dand

Z sid C1=T

sid

� � � dsi0d�1:

Next let i01 D i1 D 1 with the integral

r0X

k1D3

Z 2k1C1=T

2k1 =T� � � s�2

i1 dsi01

as before. Now assume that i2 D i1 and j2 D j1�1 D d�1. Then si2�sj2 > si2C1�sj2and let i02 D i2 C 1 and we consider

k1X

k2D0

Z sj2C2k2C1=T

sj2C2k2 =T� � � .si2C1 � sj2 /

�1 dsi02:

If i3 D i2 and j3 D j2 � 1 D d � 2, then let i03 D j3 and we consider

k1X

k3D0

Z si3�2k3 =T

si3�2k3C1=T� � � .si3 � sj3 /

�1 dsi03:

If i3 D i2 C 1 and j3 D j2, then si3 � sj3 > si3C1 � sj3 and let i03 D i3 C 1 and weconsider

k1X

k3D0

Z sj3C2k3C1=T

sj3C2k3 =T� � � .si3C1 � sj3 /

�1 dsi03:

Assuming that i2 D i1 C 1 and j2 D j1 D d, we choose i02 D i2 and estimate theintegral

k1X

k2D0

Z sj2C2k2C1=T

sj2C2k2 =T� � � .si2 � sj2 /

�1 dsi02:

If i3 D i2 and j3 D j2 � 1 D d � 1, then si3 � sj3 > si3C1 � sj3 and let i03 D i3 C 1 andwe consider

k1X

k3D0

Z sj3C2k3C1=T

sj3C2k3 =T� � � .si3C1 � sj3 /

�1 dsi03:

If i3 D i2 C 1 and j3 D j2, then let i03 D i3 and we will estimate the integral

k1X

k3D0

Z sj3C2k3C1=T

sj3C2k3 =T� � � .si3 � sj3 /

�1 dsi03:

If we have two sequences .i; j/; .k; l/ 2 I for which sid�1 � sjd�1 D sd�1 � sd <

2=T and skd�1 � sld�1 < 2=T, then let i0d D d D j1, i0d�1 D d � 1 D id�1, i0d�2 D ld�1and i01 D i1 D 1, etc. In this case we can apply the inequality .sid�2 �sjd�2 /

�1 < T=2.Continuing this process we obtain a parallelepiped P2k1 =T;:::;2kd =T with side lengths

2k1C1=T; : : : ; 2kdC1=T, where kd�1 D kd D 1. We conclude

Z

A2\B\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CTX

.il; jl/2I

r0X

k1D3

k1X

k2D0� � �

k1X

kd�2D0

�

2k1

T

��2 d�2Y

lD2

�

2kl

T

��1

Z

P2k1 =T;:::;2kd =T

j f .x � s/ � f .x/j ds

� CX

.il; jl/2I

r0X

k1D3

k1X

k2D0� � �

k1X

kd�2D02.��1=d/kkk1

2��kkk1 1ˇ

ˇP2k1 =T;:::;2kd =T

ˇ

ˇ

Z

P2k1 =T;:::;2kd =T

j f .x � s/ � f .x/j ds

� CX

.il; jl/2I

r0X

k1D3

k1X

k2D0� � �

k1X

kd�2D02.��1=d/kkk1Urf .x/ < C�:

The integrals on the sets A2 \ B \ Scr=2 and on Aj \ B . j D 3; : : : ; d � 1/ can be

estimated similarly.Now let us consider the set Bc, i.e. when sd > 1=T. Obviously, sk > 1=T .k D

1; : : : ; d � 1/. On the set A1 \ Bc we will use the inequality (5.4.33) with j D d � 1.Assume that sj1 > si1=2. Then

Z

A1\Bc\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛ X

.il; jl/2I

Z

A1\Bc\Sr=2

j f .x � s/� f .x/j s�1i1

s�˛j1

d�1Y

lD1.sil � sjl /

�1 ds

� CT�˛ X

.il; jl/2I

Z

A1\Bc\Sr=2

j f .x � s/� f .x/j s�1�˛i1

d�1Y

lD1.sil � sjl/

�1 ds:

We choose the indices i0l .l D 1 : : : ; d/ as follows. Let i01 D i1 D 1. If i2 D i1 andj2 D j1 � 1, then let i02 D j2 and if i2 D i1 C 1 and j2 D j1, then let i02 D i2. Theestimation of the integral can be finished as above.

Now assume that sj1 � si1=2. Then si1 � sj1 � si1=2 and so

Z

A1\Bc\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds (5.4.37)

� CT�˛ X

.il; jl/2I

Z

A1\Bc\Sr=2

j f .x � s/ � f .x/j s�2i1 s�˛

j1

d�1Y

lD2.sil � sjl/

�1 ds:

We choose the indices i0l .l D 1 : : : ; d/ exactly as in the integral (5.4.35). If ˛ � 1,then (5.4.37) can be estimated as above. If ˛ < 1, then we estimate (5.4.37) by

Z

A1\Bc\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT�˛ X

.il; jl/2I

Z

A1\Bc\Sr=2

j f .x � s/ � f .x/j s�1�˛i1 s�1

j1

d�1Y

lD2.sil � sjl /

�1 ds;

which can be estimated further as before. The integral on the set A1 \ Bc \ Scr=2 can

be handled similarly.On the set A2 \ Bc we will use the inequality (5.4.33) with j D d � 2. Suppose

that there exists exactly one difference sid�1 � sjd�1 for which sid�1 � sjd�1 < 2=T,say sd�1 � sd < 2=T. If sj1 > si1=2, then

Z

A2\Bc\Sr=2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds (5.4.38)

� CT1�˛X

.il; jl/2I

Z

A2\Bc\Sr=2

j f .x � s/ � f .x/j s�1i1

s�˛j1

d�2Y

lD1.sil � sjl/

�1 ds

� CT1�˛X

.il; jl/2I

Z

A2\Bc\Sr=2

j f .x � s/ � f .x/j s�1�˛i1

d�2Y

lD1.sil � sjl /

�1 ds:

Let i01 D i1, i0d D j1 D d and i0d�1 D id�1 D d � 1. We choose the remaining indicesi0l .l D 2; : : : ; d � 2/ as in the integral (5.4.36).

If sj1 � si1=2, then

Z

A2\Bc\Sr=2

j f .x � s/ � f .x/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� CT1�˛X

.il; jl/2I

Z

A2\Bc\Sr=2

j f .x � s/ � f .x/j s�2i1 s�˛

j1

d�2Y

lD2.sil � sjl /

�1 ds:

We choose the indices i0l .l D 1 : : : ; d/ exactly as before in the integral (5.4.38).The preceding integral as well as the integrals on the sets A2 \ Bc \ Sc

r=2 and onAj \ Bc . j D 3; : : : ; d � 1/ can be estimated with the ideas presented above. �


5.5 Proofs of the One-Dimensional Strong SummabilityResults

Using the results of the preceding section, now we prove the one-dimensional strongsummability results presented in Sect. 2.10.

Proof of Theorem 2.10.1 It is easy to see that

�1T

Z 1

0

� 0 t

T

�dY

jD1.stf .xj/ � f .xj// dt

D �1T

Z 1

0

� 0 t

T

�dY

jD1

�Z

R

f .xj � sj/Dt.sj/ dsj � f .xj/

�

dt

D �1T

Z 1

0

� 0 t

T

�dY

jD1lim

n!1

Z n

�n

�

f .xj � sj/ � f .xj/�

Dt.sj/ dsj dt:

Sinceˇ

ˇ

ˇ

ˇ

Z n

�nf .xj � sj/Dt.sj/ dsj

ˇ

ˇ

ˇ

ˇ

� Ct k f kW.L1;`q/

andˇ

ˇ

ˇ

ˇ

Z n

�nf .xj/Dt.sj/ dsj

ˇ

ˇ

ˇ

ˇ

D ˇ

ˇ f .xj/ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z n

�n

sin.tsj/

�sjdsj

ˇ

ˇ

ˇ

ˇ

D ˇ

ˇ f .xj/ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Z nt

�nt

sin u

�udu

ˇ

ˇ

ˇ

ˇ

� Cˇ

ˇ f .xj/ˇ

ˇ ;

we obtain

�1T

Z 1

0

� 0 t

T

�dY

jD1.stf .xj/� f .xj// dt

D limn!1

�1T

Z 1

0

� 0 t

T

�

Z

Œ�n;n�d

dY

jD1

�

f .xj � sj/ � f .xj/�

D1t .s/ ds dt

5.5 Proofs of the strong summability results 375

D limn!1

Z

Œ�n;n�d

�1T

Z 1

0

� 0 t

T

�dY

jD1

�

f .xj � sj/ � f .xj/�

D1t .s/ dt ds

DZ

Rd

dY

jD1

�

f .xj � sj/ � f .xj/�

K1;�T .s/ ds: (5.5.1)

The theorem follows from Theorem 5.4.19. �

Proof of Theorem 2.10.3 The theorem follows from the preceding proof and fromTheorem 5.4.18. �Proof of Theorem 2.10.10 for d D 2 By (5.5.1), we have to prove that

limT!1

Z

R2

j. f .x � s/ � f .x//. f .y � t/ � f .y//jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt D 0:

We are using the notation of Theorem 5.4.15. Recall that every Gabisoniya pointis a Lebesgue point. Taking into account the first part of Theorem 5.4.14 andCorollary 5.4.12, we can prove as in Theorem 5.4.15 that for i D 1; 2; 3,

Z

Ai

j. f .x � s/ � f .x//. f .y � t/ � f .y//jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt < C�

if T is large enough. The reason of this is that in the estimation of these terms wehave used U.1/

r f and M.1/f in Theorem 5.4.15. The estimations for the sets A4 andA5 cannot be used from that theorem because there we used U.2/

r f and M.2/f .On the sets A4 and A5 we decompose the integrals in another way. By (5.4.8)

with ˇ D 1,

Z

A4

j. f .x � s/ � f .x//. f .y � t/ � f .y//jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iD2

X

1� j<.iC1/=2T�˛

�

i

T

��2 � j

T

��˛

Z .iC1/=T

i=T

Z s�j=T

s�. jC1/=Tj. f .x � s/ � f .x//. f .y � t/ � f .y//j dt ds

� C1X

iD2

X

1� j<.iC1/=2T�˛

�

i

T

��2 � j

T

��˛

Z .iC1/=T

i=Tj f .x � s/� f .x/j ds

Z .iC1/=T�j=T

i=T�. jC1/=Tj f .y � t/ � f .y/j dt:

Since 2ab � a2 C b2 and ˛ > 1, we conclude

Z

A4

j. f .x � s/� f .x//. f .y � t/ � f .y//jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iD2

T

i

Z .iC1/=T

i=Tj f .x � s/ � f .x/j ds

!2

C C1X

iD2

X

1� j<.iC1/=2j�˛

T

i

Z .iC1/=T�j=T

i=T�. jC1/=Tj f .y � t/ � f .y/j dt

!2

� � C C1X

jD1

1X

iD2j

j�˛

T

i

Z .iC1/=T�j=T


!2

if T is large enough because x is a Gabisoniya point of f . In the second summand letk D i � j � 1, then k C 1 � i. Hence

C1X

jD1

1X

iD2j

j�˛

T

i

Z .iC1/=T�j=T


!2

� C1X

jD1j�˛

1X

kD0

T

k C 1

Z .kC1/=T

k=Tj f .y � t/ � f .y/j dt

!2

� C1X

iD0

T

i C 1

Z .iC1/=T

i=Tj f .y � t/ � f .y/j dt

!2

! 0;

as T ! 1. Similarly,

Z

A5

j. f .x � s/� f .x//. f .y � t/ � f .y//jˇ

ˇ

ˇK1;�T .s; t/

ˇ

ˇ

ˇ ds dt

� C1X

iD2

�

i

T

��2

Z .iC1/=T

i=T

Z s

s�1=Tj. f .x � s/ � f .x//. f .y � t/ � f .y//j dt ds

� C1X

iD2

�

i

T

��2

Z .iC1/=T

i=Tj f .x � s/� f .x/j ds

Z .iC1/=T

.i�1/=Tj f .y � t/ � f .y/j dt;


which can be estimated further by

C1X

iD2

T

i

Z .iC1/=T

i=Tj f .x � s/ � f .x/j ds

!2

C C1X

iD2

T

i

Z .iC1/=T

.i�1/=Tj f .y � t/ � f .y/j dt

!2

! 0;

as T ! 1. The proof of the theorem is complete. �Note that in this proof we have used only D 2 in the definition of the

Gabisoniya points.

Proof of Theorem 2.10.10 for d � 3 Suppose that d D 3 and use the notations ofthe proof of Theorem 5.4.15. For a fix x, set

fj.sj/ WD f0.xj � sj/� f0.xj/ . j D 1; 2; 3/

and

ai.xj/ WD T

i

Z .iC1/=T

.i�1/=T

ˇ

ˇ fj.sj/ˇ

ˇ dsj . j D 1; 2; 3; i � 1/ :

We have to prove again that

limT!1

Z

R3

j f1.s1/f2.s2/f3.s3/jˇ

ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds D 0: (5.5.2)

Since every Gabisoniya point is a Lebesgue point, we can prove as in Theo-rem 5.4.15 that (5.5.2) holds if we integrate over the sets A1, A2, A3 \ .B2 [ B3 [B4/\ C3 and A4 \ B3 \ C3.

For the remaining sets we use a decomposition method as in the proof for d D 2.By (5.4.19),

Z

A3\B1


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iD8T

�

i

T

��2 Z .iC1/=T

i=T

Z s1

s1�4=T

Z 1=T

0

j f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8T

�

i

T

��2 Z .iC1/=T

i=T

Z .iC1/=T

.i�4/=T

Z 1=T

0


� CTZ 1=T

0

j f3.s3/j ds3

1X

iD8ai.x1/ .ai.x2/C ai�2.x2/C ai�3.x2//

� CTZ 1=T

0

j f3.s3/j ds3

1X

iD1ai.x1/

2

!1=2 1X

iD1ai.x2/

2

!1=2

! 0;

as T ! 1, because xj is a Gabisoniya point. Similarly, by (5.4.21),

Z

A3\.B2[B3[B4/\C2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iD8

X

1� j<.iC1/=2T1�˛

�

i

T

��2 � j

T

��˛

Z .iC1/=T

i=T

Z s1�j=T

s1�. jC1/=T

Z 1=T

0


� C1X

iD8

X

1� j<.iC1/=2T1�˛

�

i

T

��2 � j

T

��˛

Z .iC1/=T

i=T

Z .iC1/=T�j=T

i=T�. jC1/=T

Z 1=T

0


� CTZ 1=T

0

j f3.s3/j ds3

1X

iD8

X

1� j<.iC1/=2j�˛ai.x1/ai�j.x2/:

This can be estimated further by

CTZ 1=T

0

j f3.s3/j ds3

0

@

1X

iD8

X

1� j<.iC1/=2j�˛ai.x1/

2

1

A

1=20

@

1X

iD8

X

1� j<.iC1/=2j�˛ai�j.x2/

2

1

A

1=2

� CTZ 1=T

0

j f3.s3/j ds3

1X

iD8ai.x1/

2

!1=20

@

1X

jD1j�˛

1X

iD2j

ai�j.x2/2

1

A

1=2

;

which tends to 0, as T ! 1.

On the set A4 \ B1 \ C2, we use (5.4.24) to obtain

Z

A4\B1\C2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iD8

X

1� j<.iC1/=2

�

i

T

��2 � j

T

��1

Z .iC1/=T

i=T

Z s1�j=T

s1�. jC1/=T

Z s1�s2

s1�s2�1=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8

X

1� j<.iC1/=2

�

T

i

��

T

i � j

��

T

j

�

Z .iC1/=T

i=T

Z .iC1/=T�j=T

i=T�. jC1/=T

Z . jC1/=T

. j�1/=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8ai.x1/

X

1� j<.iC1/=2ai�j.x2/aj.x3/:

Applying Hölder’s and then Young’s inequality (Theorem 1.1.7) with r D 2, p Dq D 4=3, we conclude

Z

A4\B1\C2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C

1X

iD8ai.x1/

2

!1=2

0

B

@

1X

iD8

0

@

X

1� j<.iC1/=2ai�j.x2/aj.x3/

1

A

21

C

A

1=2

� C

1X

iD8ai.x1/

2

!1=2 1X

iD1ai.x2/

4=3

!3=4 1X

iD1ai.x3/

4=3

!3=4

! 0;

as T ! 1. Similarly,

Z

A4\B1\C3


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iD8

X

1� j<.iC1/=2

�

i

T

��2 � j

T

��1

Z .iC1/=T

i=T

Z . jC1/=T

j=T

Z s1�s2

s1�s2�1=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8

X

1� j<.iC1/=2

�

T

i

��

T

i � j

��

T

j

�

Z .iC1/=T

i=T

Z . jC1/=T

j=T

Z .iC1/=T�j=T

i=T�. jC2/=Tj f1.s1/f2.s2/f3.s3/j ds:

This integral can be estimated just as before. By (5.4.26),

Z

A4\B2\C2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<. jC1/=2T�˛

�

i

T

��1 � i � j

T

��1 � j � k

T

��1 � k

T

��˛

Z .iC1/=T

i=T

Z s1�j=T

s1�. jC1/=T

Z s1�s2�k=T

s1�s2�.kC1/=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<. jC1/=2

�

T

i

��

T

i � j

��

T

j � k

�

k�˛

Z .iC1/=T

i=T

Z .iC1/=T�j=T

i=T�. jC1/=T

Z . jC1/=T�k=T

j=T�.kC1/=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<. jC1/=2ai.x1/ai�j.x2/aj�k.x3/k

�˛:

Hölder’s inequality yields that

Z

A4\B2\C2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C

1X

iD8ai.x1/

2

!1=2

0

B

@

1X

iD8

0

@

X

1� j<.iC1/=2

X

1�k<. jC1/=2ai�j.x2/aj�k.x3/k

�˛1

A

21

C

A

1=2

:

Furthermore, applying Young’s inequality twice, we obtain

Z

A4\B2\C2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C k.ai.x1//k2 k.ai.x2//k4=3

.aj.x3// � .k�˛/

4=3

� C k.ai.x1//k2 k.ai.x2//k4=3

.aj.x3//

8=7k.k�˛/k8=7 ! 0;

as T ! 1. We get in the same way that

Z

A4\B2\C3


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<.i�jC1/=2T�˛

�

i

T

��1 � j

T

��1 � i � j � k

T

��1 � k

T

��˛

Z .iC1/=T

i=T

Z . jC1/=T

j=T

Z s1�s2�k=T

s1�s2�.kC1/=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<.i�jC1/=2

�

T

i

��

T

j

��

T

i � j � k

�

k�˛

Z .iC1/=T

i=T

Z . jC1/=T

j=T

Z .i�jC1/=T�k=T

.i�j�1/=T�.kC1/=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<.i�jC1/=2ai.x1/aj.x2/ai�j�k.x3/k

�˛;

which tends to 0 as before. Finally,

Z

A4\B3\C2


ˇ

ˇK1;�T .s/

ˇ

ˇ

ˇ ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<. jC1/=2T�˛

�

i

T

��1 � i � j

T

��1 � k

T

��1 � j � k

T

��˛

Z .iC1/=T

i=T

Z s1�j=T

s1�. jC1/=T

Z .kC1/=T

k=Tj f1.s1/f2.s2/f3.s3/j ds;

which is equal to

C1X

iD8

X

1� j<.iC1/=2

X

1�k<. jC1/=2

�

T

i

��

T

i � j

�

. j � k/�˛�

T

k

�

Z .iC1/=T

i=T

Z .iC1/=T�j=T

i=T�. jC1/=T

Z .kC1/=T

k=Tj f1.s1/f2.s2/f3.s3/j ds

� C1X

iD8

X

1� j<.iC1/=2

X

1�k<. jC1/=2ai.x1/ai�j.x2/ . j � k/�˛ ak.x3/:

By Young’s inequality this tends again to 0. The proof of the theorem is complete.�

Chapter 6Rectangular Summability of Multi-DimensionalFourier Transforms

In the last chapter, we deal with the rectangular summability of Fourier transformsdefined by a function � D �1 ˝ � � � ˝ �d. We consider two types of convergence,the so-called restricted and unrestricted convergence. In the first case, T is in acone and T ! 1 while in the second case, min.T1; : : : ;Td/ ! 1, which iscalled Pringsheim’s convergence. Analogously, we consider two types of maximaloperators, the restricted and unrestricted ones. We prove similar results as inChap. 5. For the restricted convergence, we use the Hardy space H�

p .Rd/ and for

the unrestricted Hp.Rd/. We show that both maximal operators are bounded from

the corresponding Hardy space to Lp.Rd/, which implies the almost everywhere

convergence. In both cases, the set of convergence is characterized as two types ofLebesgue points.

6.1 Norm Convergence of Rectangular Summability Means

The Lp.Rd/-norm convergence of the rectangular Dirichlet integrals of the Fourier

transforms was proved in Theorem 4.3.6 for 1 < p < 1. That theorem does nothold for p D 1 and p D 1, so we consider summability methods again. In thischapter we will always assume that

� D �1 ˝ � � � ˝ �d; �j 2 L1.R/\ C0.R/ and �j.0/ D 1 (6.1.1)

for all j D 1; : : : ; d.


383

384 6 Rectangular summability

Definition 6.1.1 The Tth rectangular �-mean of the function f 2 Lp.Rd/ .1 � p �

2/ is given by

��T f .x/ WD 1

.2�/d=2

Z

Rd

dY

jD1�j

��tjTj

�

bf .t/e{x�t dt .T 2 RdC/:

As in the one-dimensional case, the integral is well defined. For an integrablefunction f ,

��T f .x/ DZ

Rdf .x � t/K�

T .t/ dt D f � K�T .x/ .x 2 R

d;T 2 RdC/:

Definition 6.1.2 The Tth rectangular �-kernel is given by

K�T .x/ WD 1

.2�/d

Z

Rd

dY

jD1�j

��tjTj

�

e{x�t dt D 1

.2�/d=2

dY

jD1Tjb�j.Tjxj/:


ˇ

ˇK�T

ˇ

ˇ � CdY

jD1Tj

�

T 2 RdC�

:

Moreover, the rectangular �-means can be rewritten as

��T f .x/ D .2�/�d=2

0

@

dY

jD1Tj

1

A

Z

Rdf .x � t/

dY

jD1b�j.Tjtj/ dt:

For b�j 2 L1.R/ . j D 1; : : : ; d/, we can extend the definition of the �-means inthe following way.

Definition 6.1.3 If b�j 2 L1.R/ for all j D 1; : : : ; d, then we extend the Tthrectangular �-mean to all f 2 W.L1; `1/.Rd/ by

��T f WD f � K�T .T 2 R

dC/:

Since

K�T D K�1

T1˝ � � � ˝ K�d

Td; (6.1.2)

6.1 Rectangular norm convergence 385

Definition 6.1.2 implies that K�T 2 L1.Rd/. Here K

�j

Tjdenotes the corresponding one-

dimensional kernel. For even and differentiable functions �j, the equality

��T f .x/ D �1Qd

jD1 Tj

Z 1

0

dY

jD1� 0

j

�

tjTj

�

stf .x/ dt

can be shown as in the one-dimensional case. Hence, for the Fejér means, i.e. if each

�j.t/ WD�

1 � jtj; if jtj � 1I0; if jtj > 1;

we obtain

��T f .x/ D 1Qd

jD1 Tj

Z T1

0

: : :

Z Td

0

stf .x/ dt:

For the two-dimensional rectangular Fejér kernel see Fig. 6.1.By (6.1.2), the following theorem can be proved exactly as Theorem 2.6.4.

−20

2

−2

0

2

0

0.5

1

Fig. 6.1 The rectangular Fejér kernel K�T with d D 2, T1 D 3, T2 D 5

Theorem 6.1.4 Assume that B is a homogeneous Banach space on Rd. If �j 2

L1.R/ and b�j 2 L1.R/ for all j D 1; : : : ; d, then

��T f

B� C k f kB

�

f 2 B;T 2 RdC�

and

limT!1 ��T f D f in the B-norm for all f 2 B:

Besides the usual spaces C0.Rd/, Lp.Rd/, W.Lp; `q/.R

d/, W.Lp; c0/.Rd/,W.C; `q/.R

d/ .1 � p; q < 1/, Cu.Rd/, the space WI.Lp; c0/.Rd/ is also a

homogeneous Banach space. Indeed, it is easy to see that Cc.Rd/ is dense in

WI.Lp; c0/.Rd/. Note that the iterated Wiener amalgam spaces WI.Lp; `q/.Rd/ were

introduced in Sect. 3.1.2.

Corollary 6.1.5 If f is a uniformly continuous and bounded function, �j 2 L1.R/

and b�j 2 L1.R/ for all j D 1; : : : ; d, then

limT!1 ��T f D f uniformly.

6.2 Almost Everywhere Restricted Summability

In this section and in the following one, we assume that T 2 Rd is in a cone. For a

given ! � 1, we define a cone by

Rd! WD fx 2 R

dC W !�1 � xi=xj � !; i; j D 1; : : : ; dg:The choice ! D 1 obviously yields the diagonal.

Definition 6.2.1 The restricted maximal operator �� is defined by

�� f WD supT2Rd

!

ˇ

ˇ��T fˇ

ˇ :

As we can see on Fig. 6.2, in the restricted maximal operator the supremumis taken on a cone only. Marcinkiewicz and Zygmund [244] were the first whoconsidered the restricted convergence. Using the condition (2.8.1) with N D 0,we show that the restricted maximal operator is bounded from H�

p .Rd/ to Lp.R

d/

(Weisz [348, 355]).

Theorem 6.2.2 For �j 2 L1.R/ suppose that b�j . j D 1; : : : ; d/ is differentiable, b�j0

is bounded and there exists 1 < ˇj < 1 such that

ˇ

ˇ

ˇ

b�j.x/ˇ

ˇ

ˇ ;ˇ

ˇ

ˇ

b�j0.x/

ˇ

ˇ

ˇ � Cjxj�ˇj .x ¤ 0; j D 1; : : : d/: (6.2.1)

6.2 Almost everywhere restricted summability 387

Fig. 6.2 The cone for d D 2

If

p1 WD max

�

d

d C 1;

1

ˇj ^ 2 ; j D 1; : : : ; d

�

 2, the verification is verysimilar. Instead of x and I1; I2 we will write .x; y/ and I; J, respectively. Let a be anarbitrary cube p-atom with support I J and

Œ�2�K�2; 2�K�2� � I; J � Œ�2�K�1; 2�K�1� .K 2 Z/:

Choose s 2 N such that 2s�1 < ! � 2s. It is easy to see that if T1 � S orT2 � S, then we have T1;T2 � S2�s. Indeed, since T is in a cone, T1 � S impliesT2 � !�1T1 � S2�s. Again, it is enough to prove that

Z

R2n4.IJ/

ˇ

ˇ��a.x; y/ˇ

ˇ

pdx dy � Cp:

First suppose that 1 < ˇj � 2 for both j D 1; 2 and let us integrate over .R n4I/ 4J. Obviously,

Z

Rn4I

Z

4J

ˇ

ˇ��a.x; y/ˇ

ˇ

pdx dy

�1X

jijD1

Z .iC1/2�K

i2�K

Z

4Jsup

T1;T2�2K�s

ˇ

ˇ��T a.x; y/ˇ

ˇ

pdx dy

C1X

jijD1

Z .iC1/2�K

i2�K

Z

4Jsup

T1;T2<2K

ˇ

ˇ��T a.x; y/ˇ

ˇ

pdx dy

DW .A/C .B/:

We may suppose that i > 0. Since b�2 2 L1.Rd/, by (6.2.1),

ˇ

ˇ��T a.x; y/ˇ

ˇ D CT1T2

ˇ

ˇ

ˇ

ˇ

Z

I

Z

Ja.t; u/b�1.T1.x � t//b�2.T2. y � u// dt du

ˇ

ˇ

ˇ

ˇ

� Cp22K=p

Z

IT1�ˇ11 jx � tj�ˇ1 dt: (6.2.2)

For x 2 Œi2�K ; .i C 1/2�K/ .i � 1/ and t 2 I, we have

jx � tj � jxj � jtj � i2�K � 2�K�1 � Ci2�K : (6.2.3)

From this, it follows that

ˇ

ˇ��T a.x; y/ˇ

ˇ � Cp22K=pCKˇ1�KT1�ˇ11 i�ˇ1 :

Since T1 � 2K2�s, we obtain

.A/ � Cp

1X

iD12�2K22KCKˇ1p�Kp2Kp�Kˇ1pi�ˇ1p � Cp

1X

iD1i�ˇ1p;

which is a convergent series if p > 1=ˇ1.To consider .B/, let I D J D .��; �/ for some � > 0 and

A1.x; v/ WDZ x

��a.t; v/ dt and A2.x; y/ WD

Z y

��A1.x; t/ dt:

Then

jAk.x; y/j � Cp2K.2=p�k/: (6.2.4)

Integrating by parts, we get that

Z

Ia.t; u/b�1.T1.x � t// dt

D A1.�; u/b�1.T1.x � �//� T1

Z

IA1.t; u/b�1

0.T1.x � t// dt: (6.2.5)

Since b�2 is bounded, for x 2 Œi2�K ; .i C 1/2�K/, (6.2.4) implies

T1T2

ˇ

ˇ

ˇ

ˇ

Z

JA1.�; u/b�1.T1.x � t//b�2.T2. y � u// du

ˇ

ˇ

ˇ

ˇ

� CpT222K=p�K2�KT1�ˇ11 jx � �j�ˇ1

� Cp22K=pCKˇ1�2KT2�ˇ11 i�ˇ1 :

Similarly,

T21T2

ˇ

ˇ

ˇ

ˇ

Z

J

Z

IA1.t; u/b�1

0.T1.x � t//b�2.T2. y � u// du dt

ˇ

ˇ

ˇ

ˇ

� Cp22K=p�K

Z

IT2�ˇ11 jx � tj�ˇ1 dt

� Cp22K=pCKˇ1�2KT2�ˇ11 i�ˇ1 :

Consequently,

.B/ � Cp

1X

iD12�2K22KCKˇ1p�2Kp2K.2�ˇ1/pi�ˇ1p � Cp

1X

iD1i�ˇ1p < 1:

Hence, we have proved that in this case

Z

Rn4I

Z

4J

ˇ

ˇ��a.x; y/ˇ

ˇ

pdx dy � Cp:

Next, we integrate over .R n 4I/ .R n 4J/,

Z

Rn4I

Z

Rn4J

ˇ

ˇ��a.x; y/ˇ

ˇ

pdx dy

�1X

jijD1

1X

jjjD1

Z .iC1/2�K

i2�K

Z . jC1/2�K

j2�Ksup

T1;T2�2K�s

ˇ

ˇ��T a.x; y/ˇ

ˇ

pdx dy

C1X

jijD1

1X

jjjD1

Z .iC1/2�K

i2�K

Z . jC1/2�K

j2�Ksup

T1;T2<2K

ˇ

ˇ��T a.x; y/ˇ

ˇ

pdx dy

DW .C/C .D/:

We may suppose again that i; j > 0.

For x 2 Œi2�K ; .i C 1/2�K/ and y 2 Œj2�K ; . j C 1/2�K/, we have by (6.2.1)and (6.2.2) that

ˇ

ˇ��T a.x; y/ˇ

ˇ � Cp22K=p

Z


Z

JT1�ˇ22 jy � uj�ˇ2 du

� Cp22K=pCKˇ1CKˇ2�2KT1�ˇ11 T1�ˇ22 i�ˇ1 j�ˇ2 :

This implies that

.C/ � Cp

1X

iD1

1X

jD12�2K22KCKˇ1pCKˇ2p�2Kp2Kp�Kˇ1p2Kp�Kˇ2pi�ˇ1pj�ˇ2p

� Cp

1X

iD1

1X

jD1i�ˇ1pj�ˇ2p < 1:

Using (6.2.5) and integrating by parts in both variables, we get that

T1T2

Z

I

Z

Ja.t; u/b�1.T1.x � t//b�2.T2. y � u// dt du

D �T1T22

Z

JA2.�; u/b�1.T1.x � �//b�2

0.T2. y � u// du

C T21T2

Z

IA2.t; �/b�1

0.T1.x � t//b�2.T2. y � �// dt

� T21T22

Z

I

Z

JA2.t; u/b�1

0.T1.x � t//b�20.T1. y � u// dt du

DW E1T.x; y/C E2T.x; y/C E3T.x; y/:

Note that A2.�;��/ D A2.�; �/ D 0. Since b�j is bounded as well, (6.2.1) implies

ˇ

ˇ

ˇ

b�j.x/ˇ

ˇ

ˇ Dˇ

ˇ

ˇ

b�j.x/ˇ

ˇ

ˇ

�jˇ

ˇ

ˇ

b�j.x/ˇ

ˇ

ˇ

1��j � Cjxj�ˇj.1��j/

for all 0 � �j � 1 and j D 1; 2. Clearly, the same is valid for b�j0. Inequalities (6.2.3)

and (6.2.4) imply

ˇ

ˇE1T.x; y/ˇ

ˇ (6.2.6)

� Cp22K=p�2KT1�ˇ1.1��1/1 jx � �j�ˇ1.1��1/

Z

JT2�ˇ2.1��2/2 jy � uj�ˇ2.1��2/ du

� Cp22K=p�3KT1�ˇ1.1��1/1 2Kˇ1.1��1/i�ˇ1.1��1/T2�ˇ2.1��2/2 2Kˇ2.1��2/j�ˇ2.1��2/;

whenever x 2 Œi2�K ; .i C 1/2�K/, y 2 Œj2�K ; . j C 1/2�K/ and 0 � �1; �2 � 1. If

1 � ˇ1.1 � �1/C 2 � ˇ2.1 � �2/ � 0;

then

supn;m<2K

ˇ

ˇE1T.x; y/ˇ

ˇ � Cp22K=pi�ˇ1.1��1/j�ˇ2.1��2/

because T is in a cone. Choosing

�j WD ˇj � 3=2ˇj

_ 0;

we can see thatZ

Rn4I

Z

Rn4Jsup

T1;T2<2K

ˇ

ˇE1T.x; y/ˇ

ˇ

pdx dy

� Cp

1X

iD1

1X

jD12�2K22Ki�.3=2^ˇ1/pj�.3=2^ˇ2/p;

which is a convergent series. The analogous estimate forˇ

ˇE2T.x; y/ˇ

ˇ can be provedsimilarly.

For x 2 Œi2�K ; .i C 1/2�K/ and y 2 Œj2�K ; . j C 1/2�K/, we conclude that

ˇ

ˇE3T.x; y/ˇ

ˇ � Cp22K=p�2K

Z


Z

JT2�ˇ22 jy � uj�ˇ2 du

� Cp22K=p�2KCKˇ1CKˇ2�2KT2�ˇ11 T2�ˇ22 i�ˇ1 j�ˇ2 :

SoZ

Rn4I

Z

Rn4Jsup

T1;T2<2K

ˇ

ˇE3T.x; y/ˇ

ˇ

pdx dy

� Cp

1X

iD1

1X

jD12�2K22K�2KpCKˇ1pCKˇ2p�2Kp24Kp�Kˇ1p�Kˇ2pi�ˇ1pj�ˇ2p

� Cp

1X

iD1

1X

jD1i�ˇ1pj�ˇ2p < 1

by the hypothesis. The integration over 4I .R n 4J/ can be done as above. Thisfinishes the proof of the theorem when 1 < ˇj � 2, j D 1; 2.

Now suppose that ˇj > 2 for some j D 1; 2. Since b�j and b�j0 are bounded and

since jxj�ˇj � jxj�2 if jxj � 1, we conclude that

ˇ

ˇ

ˇ

b�j.x/ˇ

ˇ

ˇ ;ˇ

ˇ

ˇ

b�j0.x/

ˇ

ˇ

ˇ � Cjxj�2 .x ¤ 0/; (6.2.7)


Remark 6.2.3 In the d-dimensional case, the constant d=.d C 1/ appears if weinvestigate the corresponding term to E1T . More exactly, if we integrate the term

dY

jD1Tj

Z

Id

A.�; � � � ; �; u/b�1.T1.x1 � �// � � � b�d�1.Td�1.xd�1 � �//b�d0.Td.xd � u// du

over .R n 4I1/ � � � .R n 4Id/ similar to (6.2.6), then we get that p > d=.d C 1/.Since H�

p .Rd/ � Lp.R

d/ for 1 < p � 1, we have

��f

p� Cp k f kp . f 2 Lp.R

d/; 1 < p � 1/:

As we have seen in Theorem 2.8.3, in the one-dimensional case, the Fejér operator�� is not bounded from H�

p .Rd/ to Lp.R

d/ if 0 0

��.�� f > �/ � Ck f k1:

Corollary 6.2.5 Suppose the same conditions as in Theorem 6.2.2. If f 2 Lp.Rd/

for some 1 � p < 1, then

limT!1; T2Rd

!

��T f D f a.e.

This result was proved by Marcinkiewicz and Zygmund [244] for the two-dimensional Fejér means. The general version of Corollary 6.2.5 is due to the author[348, 355]. It is easy to see that the summability methods given in Sect. 2.11 allsatisfy the conditions of Theorem 6.2.2.

6.3 Restricted convergence at Lebesgue points 393

6.3 Restricted Convergence at Lebesgue Points

Introducing the weighted Lebesgue spaces, we briefly write L�p.Rd/ .� � 0/ instead

of the L�p.Rd; �/ space equipped with the norm

k f kL�pWD�Z

Rdj f .x/.1C jxj/� jp dx

�1=p

.1 � p < 1/;

with the usual modification for p D 1. If � D 0, then we get back the Lp.Rd/

spaces. Clearly, Lp.Rd/ L�p.R

d/. We are generalizing the definition of the Herzspaces introduced in Definitions 2.7.1 and 5.4.3.


d/ is in the Herz space E�q.Rd/ .1 � q �

1; � � 0/ if

k f kE�qWD

1X

k1D0� � �

1X

kdD02kkk1.�C1�1=q/

f1Qk1��Qkd

q< 1;

where

Qk WD I.0; 2k/ n I.0; 2k�1/ .k > 0/; Q0 WD I.0; 1/:

Recall that I.0; c/ D fx 2 R W jxj < cg. It is clear that E0q.R/ D Eq.R/ andE�q.R

d/ � Lq.Rd/. Moreover,

E�q.Rd/ E�

0

q .Rd/ 0 � � < � 0 < 1

and

L1.Rd/ L�1.R

d/ D E�1.Rd/ E�q.R

d/ E�q0.Rd/ E�1.Rd/

for any 1 < q < q0 < 1 with continuous embeddings. We show that f 2 E�1.R/ ifand only if f has a decreasing majorant function belonging to L�1.R/.

Theorem 6.3.2 Let �.x/ WD supjtj�jxj j f .t/j. Then f 2 E�1.R/ if and only if � 2L�1.R/ .� � 0/ and

C�1 k�kL�1� k f kE�1

� C k�kL�1C �.0/:

Proof If � 2 L�1.R/, then

k f kE�1�

1X

kD02k.�C1/ k�1Qk k1 D

1X

kD12k.�C1/�.2k�1/C �.0/ � C k�kL�1

C �.0/:

For the converse we use the function � introduced in the proof of Theorem 2.7.3to obtain

k�kL�1� k�kL�1

D1X

kD0ank

Z

B.0;2nk /nB.0;2nk�1 /

.1C x/� dx

D C1X

kD02nk� .2nk � 2nk�1 / ank � C k f kE�1

;

which proves the theorem. �As in the previous section, let � D �1 ˝ � � � ˝ �d satisfy (6.1.1). It is easy to

see that b� 2 E�q.Rd/ if and only if b�j 2 E�q.R/ for all j D 1; : : : ; d. Note that all

examples from Sect. 2.11 satisfy this condition.

Theorem 6.3.3 Let � 2 L1.Rd/, 1 � p < 1 and 1=p C 1=q D 1. Ifb� 2 E�q.Rd/

for some � > 0, then for all f 2 Llocp .R

d/,

�� f .x/ � Cp

b�

E�qM.1/

p f .x/ .x 2 Rd/:

Proof We have

ˇ

ˇ��T f .x/ˇ

ˇ D .2�/�d=2

0

@

dY

jD1Tj

1

A

ˇ

ˇ

ˇ

ˇ

Z

Rdf .x � t/b�.T1t1; : : : ;Tdtd/ dt

ˇ

ˇ

ˇ

ˇ

� C

0

@

dY

jD1Tj

1

A

1X

k1D0� � �

1X

kdD0(6.3.1)

Z

Qk1 .T1/� � �Z

Qkd .Td/

j f .x � t/jˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt;

where

Qi.Tk/ WD I.0; 2i=Tk/ n I.0; 2i�1=Tk/ .i � 1/; Q0.Tk/ WD I.0; 1=Tk/

.k D 1; : : : ; d/. By Hölder’s inequality,

ˇ

ˇ��T f .x/ˇ

ˇ

� C

0

@

dY

jD1Tj

1

A

1X

k1D0� � �

1X

kdD0

Z

Qk1 .T1/� � �Z

Qkd .Td/

j f .x � t/jp dt

!1=p

Z

Qk1 .T1/� � �Z

Qkd .Td/

ˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ

qdt

!1=q

D C

0

@

dY

jD1Tj

1

A

1�1=q 1X

k1D0� � �

1X

kdD0

Z

Qk1

� � �Z

Qkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

Z

Qk1 .T1/� � �Z

Qkd .Td/

j f .x � t/jp dt

!1=p

: (6.3.2)

Since T 2 Rd! , we conclude

ˇ

ˇ��T f .x/ˇ

ˇ � Cp

1X

k1D0� � �

1X

kdD02kkk1.�C1=p/

Z

Qk1

� � �Z

Qkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

2��kkk1

Td1

2kkk1Cd!

Z 2k1C!=T1

�2k1C!=T1

� � �Z 2kdC!=T1

�2kdC!=T1

j f .x � t/jp dt

!1=p

� Cp

b�

E�qM.1/

p f .x/;

which shows the theorem. �Theorems 3.1.7 and 1.2.6 imply the next two corollaries.

Corollary 6.3.4 Let � 2 L1.Rd/, 1 � p < 1, p < r � 1, 1=p C 1=q D 1 and� > 0. Ifb� 2 E�q.R

d/, then

sup�>0

��.�� f > �/1=p � Cp

b�

E�qk f kp . f 2 Lp.R

d//;

�� f

r� Cr

b�

E�qk f kr . f 2 Lr.R

d//

and

��

W.Lp;1 ;`1/� Cp

b�

E�qk f kW.Lp ;`1/ . f 2 W.Lp; `1/.Rd//;

�� f

W.Lr ;`1/� Cr

b�

E�qk f kW.Lr ;`1/ . f 2 W.Lr; `1/.Rd//:

Corollary 6.3.5 If � 2 L1.Rd/, 1 � p < 1, 1=pC1=q D 1, � > 0 andb� 2 E�q.Rd/,

then for all f 2 W.Lp; c0/.Rd/

limT!1;T2Rd

!

��T f D f a.e.

Note that E�q.Rd/ E�q0.R

d/ whenever q < q0. Now we are able to prove the

convergence of ��T f at each modified Lebesgue point for all f 2 Lp.Rd/ and even

for all f 2 W.Lp; `1/.Rd/, whenever T is in a cone.

Theorem 6.3.6 Let � 2 L1.Rd/, 1 � p < 1, 1=p C 1=q D 1, � > 0 and b� 2E�q.R

d/. If f 2 W.Lp; `1/.Rd/, x is a modified p-Lebesgue point of f and M.1/p f .x/

is finite, then

limT!1;T2Rd

!

��T f .x/ D f .x/:

Proof Since x is a modified p-Lebesgue point of f , we can fix a number r < 1

such that U.1/r;p f .x/ < �. Recall that U.1/

r;p f and the modified Lebesgue points wereintroduced in Sect. 5.4.2. Let us denote by r0 the largest number i, for which r=2 �2i=T1 < r. It is easy to see that

��T f .x/ � f .x/ D0

@

dY

jD1Tj

1

A

Z

Rd. f .x � t/ � f .x//b�.T1t1; : : : ;Tdtd/ dt

and

ˇ

ˇ��T f .x/� f .x/ˇ

ˇ �0

@

dY

jD1Tj

1

A

Z

Rdj f .x � t/ � f .x/j

ˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt

� A1.x/C A2.x/;

where

A1.x/ D0

@

dY

jD1Tj

1

A

r0�!X

k1D0� � �

r0�!X

kdD0Z

Qk1 .T1/� � �Z

Qkd .Td/

j f .x � t/ � f .x/jˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt


and

A2.x/ DX

�1;:::;�d

0

@

dY

jD1Tj

1

A

1X

k�1Dr0�!C1: : :

1X

k�j Dr0�!C1

1X

k�jC1D0: : :

1X

k�d D0Z

Qk1 .T1/� � �Z

Qkd .Td/

j f .x � t/ � f .x/jˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt;

where f�1; : : : ; �dg is a permutation of f1; : : : ; dg and 1 � j � d.Similar to (6.3.2),

jA1.x/j

�0

@

dY

jD1T1�1=q

j

1

A

r0�!X

k1D0� � �

r0�!X

kdD0

Z

Qk1

� � �Z

Qkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

Z

Qk1 .T1/� � �Z

Qkd .Td/

j f .x � t/ � f .x/jp dt

!1=p

� Cp

r0�!X

k1D0� � �

r0�!X

kdD02kkk1.�C1=p/

Z

Qk1

� � �Z

Qkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

2��kkk1

Td1

2kkk1Cd!

Z 2k1C!=T1

�2k1C!=T1

� � �Z 2kdC!=T1

�2kd C!=T1


!1=p

� Cp

b�

E�qU.1/

r;p f .x/ � Cp�

b�

E�q:

We can see in the same way that

jA2.x/j � Cp

X

�1;:::;�d

1X

k�1Dr0�!C1: : :

1X

k�j Dr0�!C1

1X

k�jC1D0: : :

1X

k�d D0

2kkk1.�C1=p/

Z

Qk1

� � �Z

Qkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

2��kkk1

Td1

2kkk1Cd

Z 2k1C!=T1

�2k1C!=T1

� � �Z 2kdC!=T1

�2kd C!=T1


!1=p

:

Since M.1/p f .x/ is finite if x is a modified p-Lebesgue point of f , we have

jA2.x/j � Cp

X

�1;:::;�d

1X

k�1Dr0�!C1: : :

1X

k�j Dr0�!C1

1X

k�jC1D0: : :

1X

k�d D0

2kkk1.�C1=p/

Z

Qk1

� � �Z

Qkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

M.1/p f .x/C j f .x/j

�

:

Since r0 ! 1 as T1 ! 1 and b� 2 E�q.Rd/, we conclude that A2.x/ ! 0 as

T ! 1, which finishes the proof. �

Corollary 6.3.7 Let � 2 L1.Rd/, 1 � p < 1, 1=p C 1=q D 1 and � > 0. Ifb� 2 E�q.R

d/ and f 2 W.Lp; `1/.Rd/ is continuous at a point x, then

limT!1;T2Rd

!

��T f .x/ D f .x/:

6.4 Almost Everywhere Unrestricted Summability

Definition 6.4.1 The unrestricted maximal operator �� is defined by

�� f WD supT2Rd

C

ˇ

ˇ��T fˇ

ˇ :

In this section, instead of the Hardy space H�p .R

d/ we have to use the Hardyspace Hp.R

d/. We will first prove that the operator �� is bounded from Lp.Rd/ to

Lp.Rd/ .1 0/

and

�� f WD supT>0

ˇ

ˇ��T fˇ

ˇ :

Obviously,

ˇ

ˇ��T fˇ

ˇ � ��T j f j .T > 0/ and �� f � �� j f j:

The next result can be proved as was Theorem 2.8.1.

6.4 Almost everywhere unrestricted summability 399

Theorem 6.4.2 Suppose that � 2 L1.R/ satisfies (6.2.1) with the exponent ˇ andthatb� 0 is bounded. If 1=.ˇ ^ 2/ < p � 1, then

�� f

p� Cpk f kHp . f 2 Hp.R//:


�� f

1 � C k f k1 . f 2 L1.R//:

Let 1 < ˇ � 2 and a be an arbitrary cube p-atom with support I � R and

Œ�2�K�2; 2�K�2� � I � Œ�2�K�1; 2�K�1�:

We may assume that x > 0. Then

Z

Rn4I

ˇ

ˇ��a.x/ˇ

ˇ

pdx �

1X

iD1

Z .iC1/2�K

i2�Ksup

T�2K

ˇ

ˇ��T a.x/ˇ

ˇ

pdx

C1X

iD1

Z .iC1/2�K

i2�Ksup

T<2K

ˇ

ˇ��T a.x/ˇ

ˇ

pdx

DW .A/C .B/:

The inequality

.A/ � Cp

can be shown as in Theorem 2.8.1. To estimate .B/, observe that

��T a.x/ D TZ

Ia.t/

ˇ

ˇ

ˇ

b�.T.x � t//ˇ

ˇ

ˇ dt

D TZ

Ia.t/

ˇ

ˇ

ˇ

b�.T.x � t//ˇ

ˇ

ˇ �ˇ

ˇ

ˇ

b�.Tx/ˇ

ˇ

ˇ

�

dt:

Thus

ˇ

ˇ��T a.x/ˇ

ˇ � TZ

Ija.t/j

ˇ

ˇ

ˇ

b�.T.x � t// �b�.Tx/ˇ

ˇ

ˇ dt:

Using Lagrange’s mean value theorem, we conclude

ˇ

ˇ

ˇ

b�.T.x � t// �b�.Tx/ˇ

ˇ

ˇ Dˇ

ˇ

ˇ

b� 0.T.x � �t//ˇ

ˇ

ˇ jTtj � TjT.x � �t/j�ˇ jtj

and the proof can be finished as in Theorem 2.8.1. Because of (6.2.7), the theoremis also valid for 2 < ˇ < 1. �

Since Hp.R/ � Lp.R/ .1 < p � 1/, Theorem 6.4.2 implies that

�� f

p� Cpk f kp . f 2 Lp.R//

for 1 < p � 1. Now, we return to the higher dimensional case and verify theLp.R

d/ boundedness of �� .

Theorem 6.4.3 Suppose that �j 2 L1.R/ satisfies (6.2.1) and thatb�j0 is bounded for

each j D 1; : : : ; d. If 1 0

ˇ

ˇ

ˇ

ˇ

Z

R

Z

R

f .t; u/K�1T1.x � t/K�2

T2. y � u/ dt du

ˇ

ˇ

ˇ

ˇ

p

dx dy

�Z

R

Z

R

supT2>0

Z

R

supT1>0

ˇ

ˇ

ˇ

ˇ

Z

R

f .t; u/K�1T1.x � t/ dt

ˇ

ˇ

ˇ

ˇ

!

ˇ

ˇ

ˇK�2T2. y � u/

ˇ

ˇ

ˇ du

!p

dy dx

� Cp

Z

R

Z

R

supT1>0

ˇ

ˇ

ˇ

ˇ

Z

R

f .t; y/K�1T1.x � t/ dt

ˇ

ˇ

ˇ

ˇ

p

dx dy

� Cp

Z

R

Z

R

j f .x; y/jp dx dy;

which proves the theorem. �In the next theorem, besides the condition (6.2.1), we use its higher order version

introduced first in (2.8.1).

Theorem 6.4.4 Suppose that �j 2 L1.R/ satisfies (6.2.1), b�j is .Nj C 1/-times

differentiable for some Nj 2 N and b�j0 is bounded for each j D 1; : : : ; d. Moreover,

assume that there exists Nj C 1 < ˇj � Nj C 2 such that

ˇ

ˇ

ˇ

ˇ

b�j

�.kj/

.x/

ˇ

ˇ

ˇ

ˇ

� Cjxj�ˇj .x ¤ 0/; (6.4.1)

whenever kj D Nj and kj D Nj C 1. If

p2 WD max

�

1

ˇj; j D 1; : : : ; d

�

< p � 1;

then

�� f

p � Cpk f kHp . f 2 Hp.Rd//: (6.4.2)

Proof By the first condition of the theorem and by Theorem 6.4.3, inequality (6.4.2)holds for 1 < p � 1. We sketch the proof by giving only the main ideas. We willprove the theorem only for d D 3, because the proof is similar for larger d or ford D 2. Choose a simple p-atom a with support R D I1 I2 A where I1 and I2 areintervals with

2�Ki�1 < jIij � 2�Ki .Ki 2 Z; i D 1; 2/

and

Œ�2�Ki�2; 2�Ki�2� � Ii � Œ�2�Ki�1; 2�Ki�1�:

We assume that ri � 2 are arbitrary integers. By Theorems 3.6.9 and 3.6.12, it isenough to show that

Z

.Ir11 /

c

Z

.Ir22 /

c

Z

A

ˇ

ˇ��.x/ˇ

ˇ

pdx � Cp2

��1r12��2r2 ; (6.4.3)

and, if A D I3 is also an interval,

Z

.Ir11 /

c

Z

.Ir22 /

c

Z

.I3/c

ˇ

ˇ�� .x/ˇ

ˇ

pdx � Cp2

��1r12��2r2 (6.4.4)

for all p2 0

ˇ

ˇ��T aˇ

ˇC supT1�2K1 ;T2<2K2

T3>0

ˇ

ˇ��T aˇ

ˇ

C supT1<2K1 ;T2�2K2

T3>0

ˇ

ˇ��T aˇ

ˇC supT1�2K1 ;T2�2K2

T3>0

ˇ

ˇ��T aˇ

ˇ : (6.4.5)

We will investigate only the second term. Obviously,

Z

.Ir11 /

c

Z

.Ir22 /

c

Z

Asup

T1�2K1 ;T2<2K2

T3>0

ˇ

ˇ��T a.x/ˇ

ˇ

pdx

�1X

ji1jD2r1�2

1X

ji2jD2r2�2

Z .i1C1/2�K1

i12�K1

Z .i2C1/2�K2

i22�K2

Z

Asup

T1�2K1 ;T2<2K2

T3>0

ˇ

ˇ��T a.x/ˇ

ˇ

pdx;

where we may suppose that il > 0. As we noted in Sect. 3.4, we may suppose that thecancellation property (iii) of the definition of the simple atoms (Definition 3.6.11)holds for all k � M, where M � M. p/ is arbitrary. So assume that M � Nj C 2 forall j D 1; 2; 3. Using Taylor’s formula

g.t/ DN�1X

kD0

g.k/.0/

kŠtk C g.N/.�t/

NŠtN

for gj.t/ D b�j.Tj.xj � tj//, N D N1;N2 C 1, where 0 < �j < 1, we conclude

��T a.x/ D T1T2T3.2�/3=2

Z

Ra.t/b�1.T1.x1 � t1//b�2.T2.x2 � t2//b�3.T3.x3 � t3// dt

D T1T2T3.2�/3=2

Z

Ra.t/

b�1.T1.x1 � t1// �N1�1X

kD0

g.k/1 .0/

kŠtk

!

b�2.T2.x2 � t2//�N2X

lD0

g.l/2 .0/

lŠtl

!

b�3.T3.x3 � t3// dt

D CT1T2

Z

Ra.t/.�1/N1CN2C1TN1

1

b�1

�.N1/.T1.x1 � �1t1//t

N11

TN2C12

b�2

�.N2C1/.T2.x2 � �2t2//t

N2C12 K�3

T3.x3 � t3/ dt:

Then by (6.4.1),

ˇ

ˇ��T a.x/ˇ

ˇ � CTN1C11 TN2C2

2

Z

I1

Z

I2

jT1.x1 � �1t1/j�ˇ1 jt1jN1 jt2jN2C1

jT2.x2 � �2t2/j�ˇ2ˇ

ˇ

ˇ

ˇ

Z

Aa.t/K�

T3 .x3 � t3/ dt3

ˇ

ˇ

ˇ

ˇ

dt1 dt2:

Since T1 � 2K1 , T2 < 2K2 and for xj 2 Œij2�Kj ; .ij C 1/2�Kj/ and tj 2Œ�2�Kl�1; 2�Kj�1/ . j D 1; 2/,

jxj � �jtjj � jxjj � jtjj � ij2�Kj � 2�Kj�1 � Cij2

�Kj ;

we have

ˇ

ˇ��T a.x/ˇ

ˇ � CTN1C1�ˇ11 TN2C2�ˇ2

2 2�K1.N1�ˇ1/2�K2.N2C1�ˇ2/i�ˇ11 i�ˇ22

Z

I1

Z

I2

ˇ

ˇ

ˇ

ˇ

Z

Aa.t/K�

T3 .x3 � t3/ dt3

ˇ

ˇ

ˇ

ˇ

dt1 dt2

� C2K1CK2 i�ˇ11 i�ˇ22

Z

I1

Z

I2

ˇ

ˇ

ˇ

ˇ

Z

Aa.t/K�

T3 .x3 � t3/ dt3

ˇ

ˇ

ˇ

ˇ

dt1 dt2:

Hence, by Hölder’s inequality,

Z

.Ir11 /

c

Z

.Ir22 /

c

Z

Asup

T1�2K1 ;T2<2K2

T3>0

ˇ

ˇ��T a.x/ˇ

ˇ

pdx

� Cp

1X

i1D2r1�2

1X

i2D2r2�2

2�K1�K22K1pCK2pi�ˇ1p1 i�ˇ2p

2

Z

A

Z

I1

Z

I2

supT3>0

ˇ

ˇ

ˇ

ˇ

Z

Aa.t/K�

T3.x3 � t3/ dt3

ˇ

ˇ

ˇ

ˇ

dt1 dt2

!p

dx3

� CpjAj1�p1X

i1D2r1�2

1X

i2D2r2�2


2

Z

A

Z

I1

Z

I2

supT3>0

ˇ

ˇ

ˇ

ˇ

Z

Aa.t/K�

T3 .x3 � t3/ dt3

ˇ

ˇ

ˇ

ˇ

dt1 dt2 dx3

!p

:

Using again Hölder’s inequality and the fact that �� is bounded on L2.Rd/ for alld � 1, we conclude

Z

.Ir11 /

c

Z

.Ir22 /

c

Z

Asup

T1�2K1 ;T2<2K2

T3>0

ˇ

ˇ��T a.x/ˇ

ˇ

pdx

� CpjAj1�p=21X

i1D2r1�2

1X

i2D2r2�2


2

0

@

Z

I1

Z

I2

Z

R

supT3>0

ˇ

ˇ

ˇ

ˇ

Z

Aa.t/K�

T3.x3 � t3/ dt3

ˇ

ˇ

ˇ

ˇ

2

dx3

!1=2

dt1 dt2

1

A

p

� CpjAj1�p=21X

i1D2r1�2

1X

i2D2r2�2


2

Z

I1

Z

I2

�Z

R

ja.t1; t2; x3/j2 dx3

�1=2

dt1 dt2

!p

� CpjAj1�p=21X

i1D2r1�2

1X

i2D2r2�2

2K1p=2CK2p=22�K1�K2 i�ˇ1p1 i�ˇ2p

2

�Z

I1

Z

I2

Z

R

ja.t/j2 dt

�p=2

:

The following inequality follows from kak2 � �

2�K1�K2 jAj�1=2�1=p:

Z

.Ir11 /

c

Z

.Ir22 /

c

Z

Asup

T1�2K1 ;T2<2K2

T3>0

ˇ

ˇ��T a.x/ˇ

ˇ

pdx � Cp

1X

i1D2r1�2

1X

i2D2r2�2

i�ˇ1p1 i�ˇ2p

2

� Cp2�r1.ˇ1p�1/2�r2.ˇ2p�1/:

The other terms of (6.4.5) can be handled in the same way, which shows (6.4.3).Obviously, the same ideas show (6.4.4). �

The next corollary follows from Corollary 3.5.8 and Theorem 3.3.2 in the usualway.

Corollary 6.4.5 Under the conditions of Theorem 6.4.4, if f 2 Hi1.R

d/ for somei D 1; : : : ; d, then

sup�>0

��.�� f > �/ � Ck f kHi1:

By the density argument, we get here almost everywhere convergence forfunctions from the spaces Hi

1.Rd/ instead of L1.Rd/. In some sense, the Hardy space

Hi1.R

d/ plays the role of L1.Rd/ in higher dimensions.

Corollary 6.4.6 Under the conditions of Theorem 6.4.4, if f 2 Hi1.R

d/ for somei D 1; : : : ; d or f 2 Lp.R

d/ for some 1 < p < 1, then


The almost everywhere convergence is not true for all f 2 L1.Rd/.A counterexample, which shows that the almost everywhere convergence is not

true for all integrable functions, is due to Gát [132]. Recall that

L1.Rd/ Hi

1.Rd/ L.log L/d�1.Rd/ Lp.R

d/ .1 < p � 1/:

6.5 Unrestricted Convergence at Lebesgue Points

First we introduce the homogeneous version of the Herz space of Definition 6.3.1.


d/ is in the homogeneous Herz space PE�q.Rd/

.1 � q � 1; � � 0/ if

k f k PE�q WD1X

k1D�1� � �

1X

kdD�12kkk1.�C1�1=q/

f1Pk1��Pkd

q< 1;

where

Pk WD I.0; 2k/ n I.0; 2k�1/ .k 2 Z/:

6.5 Unrestricted Convergence at Lebesgue Points 405

It can be shown easily that PE0q.R/ D PEq.R/ and

E�q.Rd/ � Lq.R

d/ \ PE�q.R/; k f k PE�q � Cq k f kE�q; k f kq � Cq k f kE�q

:

In the one-dimensional case, E�q.R/ D Lq.R/\ PE�q.R/. Furthermore,

L1.R/ L�1.R/ D PE�1.R/ PE�q.R/ PE�q0.R/ PE�1.R/

for any 1 � q < q0 � 1 with continuous embeddings. Similar to Theorem 6.3.2,f 2 PE�1.R/ if and only if f has a decreasing majorant function belonging to L�1.R/.

Theorem 6.5.2 Let �.x/ WD supjtj�jxj j f .t/j. Then f 2 PE�1.R/ if and only if � 2L�1.R/ .� � 0/ and

C�1 k�kL�1� k f k PE�1 � C k�kL�1

:

For � D �1 ˝ � � � ˝ �d,b� 2 PE�q.Rd/ if and only if b�j 2 PE�q.R/ for all j D 1; : : : ; d.In this section we use the strong maximal function which was introduced by

Ms;pf .x/ WD supx2I

�

1

jIjZ

Ij f jp d�

�1=p

.x 2 Rd/

in Sect. 3.1.2, where the supremum is taken over all rectangles with sides parallel tothe axes.

Theorem 6.5.3 Let � 2 L1.Rd/, 1 � p < 1 and 1=p C 1=q D 1. Ifb� 2 PE0q.Rd/,then for all f 2 Lloc

p .Rd/,

�� f .x/ � Cp

b�

PE0qMs;pf .x/:

Proof Similar to (6.3.1) and (6.3.2),

ˇ

ˇ��T f .x/ˇ

ˇ � C

0

@

dY

jD1Tj

1

A

1X

k1D�1� � �

1X

kdD�1Z

Pk1 .T1/� � �Z

Pkd .Td/

j f .x � t/jˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt

� C

0

@

dY

jD1Tj

1

A

1X

k1D�1� � �

1X

kdD�1

Z

Pk1 .T1/� � �Z

Pkd .Td/

ˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ

qdt

!1=q


Z

Pk1 .T1/� � �Z

Pkd .Td/

j f .x � t/jp dt

!1=p

D C

0

@

dY

jD1T1�1=q

j

1

A

1X

k1D�1� � �

1X

kdD�1

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

Z

Pk1 .T1/� � �Z

Pkd .Td/

j f .x � t/jp dt

!1=p

; (6.5.1)

where

Pi.Tj/ WD I.0; 2i=Tj/ n I.0; 2i�1=Tj/ .i � 1/:

If we define

G.u/ WD�Z u1

�u1

� � �Z ud

�ud

j f .x � t/jp dt

�1=p

.u D .u1; : : : ; ud/ 2 RdC/;

then

Gp.u/

2dQd

jD1 uj

� Mps;pf .x/ .u D .u1; : : : ; ud/ 2 R

dC/:

Hence

ˇ

ˇ��T f .x/ˇ

ˇ � C

0

@

dY

jD1T1=p

j

1

A

1X

k1D�1� � �

1X

kdD�1

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

G

�

2k1

T1; : : : ;

2kd

Td

�

� Cp

1X

k1D�1� � �

1X

kdD�1

0

@

dY

jD12kj=p

1

A

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

Ms;pf .x/

� Cp

b�

PE0qMs;pf .x/:

�

Taking into account Corollary 3.1.16, we obtain

Corollary 6.5.4 Let � 2 L1.Rd/, 1 � p < 1, 1=p C 1=q D 1 and I D I1 � � � Id

with jI1j D � � � D jIdj D 1. Ifb� 2 PE0q.Rd/, then

sup�>0

��.x W ��.x/ > �; x 2 I/1=p � Cp

b�

PE0q

1C k f kLp.log L/d�1

�

:

For p < r � 1,

�� f

r� Cr

b�

PE0qk f kr . f 2 Lr.R

d//:

If f 2 WI.Lp.log L/d�1; `1/.Rd/, then

�� f

W.Lp;1 ;`1/� Cp

b�

PE0q

1C k f kWI .Lp.log L/d�1;`1/

�

and, for p < r � 1,

�� f

W.Lr ;`1/� Cr

b�

PE0qk f kWI .Lr ;`1/ . f 2 WI.Lr; `1/.Rd//:

The next corollary follows from the density argument and from fact that Cc.Rd/

is dense in WI.Lp.log L/d�1; c0/.Rd/.

Corollary 6.5.5 Let � 2 L1.Rd/, 1 � p < 1 and 1=p C 1=q D 1. Ifb� 2 PE0q.Rd/,then


for all f 2 WI.Lp.log L/d�1; c0/.Rd/.Note that

C0.Rd/ � WI.Lp.log L/d�1; c0/.Rd/;

Lr.Rd/ � WI.Lr; c0/.R

d/ � WI.Lp.log L/d�1; c0/.Rd/

for 1 � p < r � 1 and

Lp.log L/d�1.Rd/ � WI.Lp.log L/d�1; c0/.Rd/ � WI.Lp; c0/.Rd/;

Lp.log L/d�1.Rd/ � Lp.Rd/ � WI.Lp; c0/.R

d/

for 1 � p < 1. The last convergence result do not hold for all f 2 Lp.Rd/. This

was proved in a special case, for p D 1 and for the Fejér means of Fourier series byGát [132] (for Walsh-Fourier series see [130]).

Theorem 3.1.15 implies

limh!0

1Qd

jD1.2hj/

Z h1

�h1

: : :

Z hd

�hd

f .x � u/ du D f .x/

for almost every x 2 Rd, where f 2 WI.L1.log L/d�1; `1/.Rd/.

Definition 6.5.6 A point x 2 Rd is called a strong p-Lebesgue point .1 � p < 1/

of f if

limh!0

1Qd

jD1.2hj/

Z h1

�h1

: : :

Z hd

�hd

j f .x � u/ � f .x/jp du

!1=p

D 0:

The following facts can be proved in the usual way. If p < r, then all strongr-Lebesgue points are strong p-Lebesgue points.

Theorem 6.5.7 Almost every point x 2 Rd is a strong p-Lebesgue point of f 2

WI.Lp.log L/d�1; `1/.Rd/ .1 � p < 1/.Now we can extend Corollary 6.5.5 to the space WI.Lp.log L/d�1; `1/.Rd/ as

follows. Note that for f 2 WI.Lp.log L/d�1; `1/.Rd/, Ms;pf is almost everywherefinite by Corollary 3.1.16.

Theorem 6.5.8 Let � 2 L1.Rd/, 1 � p < 1, 1=p C 1=q D 1 and b� 2 PE0q.Rd/.If f 2 WI.Lp.log L/d�1; `1/.Rd/, x is a strong Lebesgue point of f and Ms;pf .x/ isfinite, then


Proof Let

G.u/ WD�Z u1

�u1

� � �Z ud

�ud


�1=p

;

where u D .u1; : : : ; ud/ 2 RdC. Since x is a strong p-Lebesgue point of f , for all

� > 0 there exists m 2 Z such that

Gp.u/

2dQd

jD1 uj

� �p for 0 < uj � 2m; j D 1; : : : ; d: (6.5.2)


As we have seen before

ˇ

ˇ��T f .x/� f .x/ˇ

ˇ �0

@

dY

jD1Tj

1

A

Z

Rdj f .x � t/ � f .x/j

ˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt

� A1.x/C A2.x/;

where

A1.x/ D0

@

dY

jD1Tj

1

A

mCblog2 T1cX

k1D�1� � �

mCblog2 TdcX

kdD�1Z

Pk1 .T1/� � �Z

Pkd .Td/

j f .x � t/ � f .x/jˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt

and

A2.x/ DX

�1;:::;�d

0

@

dY

jD1Tj

1

A

1X

k�1DmCblog2 T�1 cC1: : :

1X

k�j DmCblog2 T�j cC11X

k�jC1D�1

: : :

1X

k�d D�1Z

Pk1 .T1/� � �Z

Pkd .Td/

j f .x � t/ � f .x/jˇ

ˇ

ˇ

b�.T1t1; : : : ;Tdtd/ˇ

ˇ

ˇ dt;

where f�1; : : : ; �dg is a permutation of f1; : : : ; dg and 1 � j � d.As in (6.5.1),

jA1.x/j

�0

@

dY

jD1T1�1=q

j

1

A

mCblog2 T1cX

k1D�1� � �

mCblog2 TdcX

kdD�1

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

Z

Pk1 .T1/� � �Z

Pkd .Td/


!1=p

�0

@

dY

jD1T1=p

j

1

A

mCblog2 T1cX

k1D�1� � �

mCblog2 TdcX

kdD�1

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

G

�

2k1

T1; : : : ;

2kd

Td

�

:


Inequality (6.5.2) and 2kj=Tj � 2mTj=Tj � 2m imply

jA1.x/j � Cp�

mCblog2 T1cX

k1D�1� � �

mCblog2 TdcX

kdD�10

@

dY

jD12kj=p

1

A

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

� Cp�

b�

PE0q:

Similarly,

jA2.x/j �X

�1;:::;�d

0

@

dY

jD1T1=p

j

1

A

1X


1X

k�j DmCblog2 T�j cC1

1X

k�jC1D�1

: : :

1X

k�d D�1

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

Z

Pk1 .T1/� � �Z

Pkd .Td/


!1=p

:

It is supposed that Ms;pf .x/ is finite and x is a strong p-Lebesgue point of f , so wehave

Z

Pk1 .T1/� � �Z

Pkd .Td/


� Cp

0

@

dY

jD1

2kj

Tj

1

A

Mps;p f .x/C j f .x/jp

�

;

thus

jA2.x/j � Cp

X

�1;:::;�d

1X


1X

k�j DmCblog2 T�j cC1

1X

k�jC1D�1

: : :

1X

k�d D�1

0

@

dY

jD12kj=p

1

A

Z

Pk1

� � �Z

Pkd

ˇ

ˇ

ˇ

b�.t/ˇ

ˇ

ˇ

qdt

!1=q

Ms;pf .x/C j f .x/j�

:

Since blog2 T�j c ! 1 as T ! 1 andb� 2 PE0q.Rd/, we conclude that A2.x/ ! 0 asT ! 1. �Corollary 6.5.9 Let � 2 L1.Rd/, 1 � p < 1, 1=p C 1=q D 1 andb� 2 PE0q.Rd/. Iff 2 WI.Lp.log L/d�1; `1/.Rd/ is continuous at a point x and Ms;pf .x/ is finite, then


Finally, we note again that WI.Lp.log L/d�1; `1/.Rd/ contains the spacesLp.log L/d�1.Rd/, C0.Rd/ and Lr.R

d/ for all 1 � p < r � 1.

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Index

Hp-atom, 157`q-� -kernel, 230`q-� -mean, 230, 234`q-Dirichlet integral, 213, 217`q-Dirichlet kernel, 205, 213`q-partial sum, 205� -kernel, 96� -mean, 96, 97p-Lebesgue point, 113, 317p-atom, 157. p; q/-atom, 156.p; q/-atom, 46p-atom, 40, 156

Abel summation, 132adjacent intervals, 122atomic decomposition, 30, 40, 156

ball . p; q/-atom, 156ball p-atom, 156Bessel function, 223Bessel summation, 132, 257beta function, 222Bochner-Riesz summation, 258bounded overlapping property, 33bounded tempered distribution, 21

Carleson’s theorem, 89, 93, 210, 217centered maximal function, 9circular � -means, 230circular Dirichlet integral, 213

circular Fejér kernel, 230circular partial sum, 205circular summability, 230cone, 386conjugate function, 87conjugate index, 6convolution, 7, 20cube . p; q/-atom, 156cube p-atom, 157cubic � -means, 230cubic Dirichlet integral, 213cubic Fejér kernel, 230cubic partial sum, 205cubic summability, 230

de La Vallée-Poussin summation, 132dilation, 73, 83, 204Dirac measure, 18Dirichlet integral, 91, 92Dirichlet kernel, 86, 91distance, 32distribution function, 9divided difference, 218dyadic interval, 157dyadic rectangle, 157

Fejér kernel, 97, 230, 385Fejér means, 385Fejér summation, 97, 132Fourier coefficient, 85, 204Fourier series, 85, 204Fourier transform, 72, 77, 80, 81, 83, 203


433

434 Index

Gabisoniya point, 121gamma function, 221grand maximal function, 23

Hardy spaces, 22, 152Hardy’s inequality, 50Hardy-Littlewood maximal function, 8, 138,

139, 317Hardy-Littlewood theorem, 10, 139Hardy-Lorentz spaces, 48Hausdorff-Young theorem, 81Herz space, 101, 318, 393homogeneous Banach space, 99, 234homogeneous Herz space, 102, 318, 404hybrid Hardy spaces, 152hybrid maximal functions, 152

integer part, 17interpolating function, 48interpolation space, 49inverse Fourier transform, 75, 83, 203inversion formula, 75, 78, 83, 204involution function, 73iterated Wiener amalgam space, 142

Jackson-de La Vallée-Poussin summation, 132Journé’s covering lemma, 185

Lebesgue measure, 4Lebesgue point, 113, 317, 408Lebesgue’s differentiation theorem, 14, 139local Hardy-Littlewood maximal function, 105local strong maximal function, 147locally bounded, 120, 323Lorentz space, 47Lusin area integrals, 153

Marcinkiewicz summability, 230Marcinkiewicz-Zygmund theorem, 13maximal operator, 13, 103, 210, 217, 264Minkowski’s inequality, 5modified p-Lebesgue point, 321modified Lebesgue point, 321, 396modified maximal function, 141, 142modified strong p-Lebesgue point, 321modified strong Lebesgue point, 321modulation, 73, 83

non-increasing rearrangement, 47non-tangential maximal function, 23

partial sum, 86perfect set, 121Picard summation, 132, 257Plancherel theorem, 77Poisson kernel, 22, 152Pringsheim convergence, 208

quasilinear, 49

radial function, 226rectangle p-atom, 184rectangular � -kernel, 384rectangular � -mean, 384rectangular Dirichlet integral, 213, 217rectangular Dirichlet kernel, 207, 213rectangular Fejér kernel, 385rectangular partial sum, 207reflection, 20Reiteration theorem, 50restricted maximal operator, 386Riemann-Lebesgue lemma, 80Riesz projection, 87Riesz summation, 133, 258Rogosinski summation, 132

Schwartz function, 16, 151simple p-atom, 184, 193strong p-Lebesgue point, 408strong Lebesgue point, 408strong maximal function, 143sublinear, 49

tempered distribution, 18translation, 20, 73, 83triangular � -means, 230triangular Dirichlet integral, 213triangular Fejér kernel, 230triangular partial sum, 205triangular summability, 230trigonometric polynomial, 87, 208trigonometric system, 204

unrestricted maximal operator, 398

Index 435

Vitali covering lemma, 9Vitali-Wiener covering lemma, 31

weak Lp.R/ space, 5weak Hardy spaces, 22, 152weak type .1; 1/, 10Weierstrass summation, 132, 257

weighted Lebesgue spaces, 393Whitney covering lemma, 32Wiener algebra, 98Wiener amalgam space, 98, 138

Young’s inequality, 7

convergence and summability of fourier transforms and hardy spaces

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